## Πέμπτη, 29 Δεκεμβρίου 2011

### McCay Cubic

Let ABC be a triangle, P a point and A1B1C1 the circumcevian triangle of P.

The perpendiculars from A1,B1,C1 to the sidelines of ABC, BC,CA,AB resp. intersect again the circumcircle at A2,B2,C2, resp. (other than A1,B1,C1).

Denote:

Ab := the orthogonal projection of A2 on B1B2
Ac := the orthogonal projection of A2 on C1C2

(O1) := the circumcircle of A2AbAc.

Similarly (O2), (O3).

Which is the locus of P such that (O1),(O2),(O3) are concurrent?

APH, 29 December 2011

-------------------------------------------------------------

The locus is Line at infinity + Circumcircle + McCay cubic.

If ABC is right angled
A = pi/2 then there is no locus.
For every point P the circles are concurrent at the intersection of lines B1B2, C1C2.

## Τετάρτη, 28 Δεκεμβρίου 2011

### Euler Lines Locus

Let ABC be a triangle, P a point and A'B'C' the circumcevian triangle of P.

Denote:

Ab := the orthogonal projection of A' on BB'
Ac: = the orthogonal projection of A' on CC'

L1 := the Euler Line of A'AbAc.
Similarly the lines L2,L3.

Which is the locus of P such that the L1,L2,L3 are concurrent?

ΑΠΧ, Hyacinthos #20600

The locus is the circumcircle and the quintic CYCLIC SUM {x^3*S_A*[(cy)^2-(bz)^2]}=0

This is the Euler-Morley quintic Q003
Bernard Gibert, Hyacinthos #20603

---------------------

For P = O, the lines bound a triangle which seems to be parallelogic
with the Orthic triangle.
Parallelogic Centers? (The one lies on the NPC)

ΑΠΧ, Hyacinthos #20600

For P=O, the parallelogic center on NPC is X128.

The other center is

{-4 a^22 + 24 a^20 b^2 - 62 a^18 b^4 + 94 a^16 b^6 - 99 a^14 b^8 +
77 a^12 b^10 - 35 a^10 b^12 - 3 a^8 b^14 + 11 a^6 b^16 - a^4 b^18 -
3 a^2 b^20 + b^22 + 24 a^20 c^2 - 108 a^18 b^2 c^2 +
194 a^16 b^4 c^2 - 172 a^14 b^6 c^2 + 67 a^12 b^8 c^2 -
2 a^10 b^10 c^2 + 13 a^8 b^12 c^2 - 24 a^6 b^14 c^2 -
3 a^4 b^16 c^2 + 18 a^2 b^18 c^2 - 7 b^20 c^2 - 62 a^18 c^4 +
194 a^16 b^2 c^4 - 230 a^14 b^4 c^4 + 120 a^12 b^6 c^4 -
7 a^10 b^8 c^4 - 29 a^8 b^10 c^4 + 20 a^6 b^12 c^4 +
14 a^4 b^14 c^4 - 41 a^2 b^16 c^4 + 21 b^18 c^4 + 94 a^16 c^6 -
172 a^14 b^2 c^6 + 120 a^12 b^4 c^6 - 56 a^10 b^6 c^6 +
19 a^8 b^8 c^6 + 8 a^6 b^10 c^6 - 14 a^4 b^12 c^6 +
36 a^2 b^14 c^6 - 35 b^16 c^6 - 99 a^14 c^8 + 67 a^12 b^2 c^8 -
7 a^10 b^4 c^8 + 19 a^8 b^6 c^8 - 30 a^6 b^8 c^8 + 4 a^4 b^10 c^8 +
12 a^2 b^12 c^8 + 34 b^14 c^8 + 77 a^12 c^10 - 2 a^10 b^2 c^10 -
29 a^8 b^4 c^10 + 8 a^6 b^6 c^10 + 4 a^4 b^8 c^10 -
44 a^2 b^10 c^10 - 14 b^12 c^10 - 35 a^10 c^12 + 13 a^8 b^2 c^12 +
20 a^6 b^4 c^12 - 14 a^4 b^6 c^12 + 12 a^2 b^8 c^12 -
14 b^10 c^12 - 3 a^8 c^14 - 24 a^6 b^2 c^14 + 14 a^4 b^4 c^14 +
36 a^2 b^6 c^14 + 34 b^8 c^14 + 11 a^6 c^16 - 3 a^4 b^2 c^16 -
41 a^2 b^4 c^16 - 35 b^6 c^16 - a^4 c^18 + 18 a^2 b^2 c^18 +
21 b^4 c^18 - 3 a^2 c^20 - 7 b^2 c^20 + c^22,
a^22 - 3 a^20 b^2 - a^18 b^4 + 11 a^16 b^6 - 3 a^14 b^8 -
35 a^12 b^10 + 77 a^10 b^12 - 99 a^8 b^14 + 94 a^6 b^16 -
62 a^4 b^18 + 24 a^2 b^20 - 4 b^22 - 7 a^20 c^2 + 18 a^18 b^2 c^2 -
3 a^16 b^4 c^2 - 24 a^14 b^6 c^2 + 13 a^12 b^8 c^2 -
2 a^10 b^10 c^2 + 67 a^8 b^12 c^2 - 172 a^6 b^14 c^2 +
194 a^4 b^16 c^2 - 108 a^2 b^18 c^2 + 24 b^20 c^2 + 21 a^18 c^4 -
41 a^16 b^2 c^4 + 14 a^14 b^4 c^4 + 20 a^12 b^6 c^4 -
29 a^10 b^8 c^4 - 7 a^8 b^10 c^4 + 120 a^6 b^12 c^4 -
230 a^4 b^14 c^4 + 194 a^2 b^16 c^4 - 62 b^18 c^4 - 35 a^16 c^6 +
36 a^14 b^2 c^6 - 14 a^12 b^4 c^6 + 8 a^10 b^6 c^6 +
19 a^8 b^8 c^6 - 56 a^6 b^10 c^6 + 120 a^4 b^12 c^6 -
172 a^2 b^14 c^6 + 94 b^16 c^6 + 34 a^14 c^8 + 12 a^12 b^2 c^8 +
4 a^10 b^4 c^8 - 30 a^8 b^6 c^8 + 19 a^6 b^8 c^8 - 7 a^4 b^10 c^8 +
67 a^2 b^12 c^8 - 99 b^14 c^8 - 14 a^12 c^10 - 44 a^10 b^2 c^10 +
4 a^8 b^4 c^10 + 8 a^6 b^6 c^10 - 29 a^4 b^8 c^10 -
2 a^2 b^10 c^10 + 77 b^12 c^10 - 14 a^10 c^12 + 12 a^8 b^2 c^12 -
14 a^6 b^4 c^12 + 20 a^4 b^6 c^12 + 13 a^2 b^8 c^12 -
35 b^10 c^12 + 34 a^8 c^14 + 36 a^6 b^2 c^14 + 14 a^4 b^4 c^14 -
24 a^2 b^6 c^14 - 3 b^8 c^14 - 35 a^6 c^16 - 41 a^4 b^2 c^16 -
3 a^2 b^4 c^16 + 11 b^6 c^16 + 21 a^4 c^18 + 18 a^2 b^2 c^18 -
b^4 c^18 - 7 a^2 c^20 - 3 b^2 c^20 + c^22,
a^22 - 7 a^20 b^2 + 21 a^18 b^4 - 35 a^16 b^6 + 34 a^14 b^8 -
14 a^12 b^10 - 14 a^10 b^12 + 34 a^8 b^14 - 35 a^6 b^16 +
21 a^4 b^18 - 7 a^2 b^20 + b^22 - 3 a^20 c^2 + 18 a^18 b^2 c^2 -
41 a^16 b^4 c^2 + 36 a^14 b^6 c^2 + 12 a^12 b^8 c^2 -
44 a^10 b^10 c^2 + 12 a^8 b^12 c^2 + 36 a^6 b^14 c^2 -
41 a^4 b^16 c^2 + 18 a^2 b^18 c^2 - 3 b^20 c^2 - a^18 c^4 -
3 a^16 b^2 c^4 + 14 a^14 b^4 c^4 - 14 a^12 b^6 c^4 +
4 a^10 b^8 c^4 + 4 a^8 b^10 c^4 - 14 a^6 b^12 c^4 +
14 a^4 b^14 c^4 - 3 a^2 b^16 c^4 - b^18 c^4 + 11 a^16 c^6 -
24 a^14 b^2 c^6 + 20 a^12 b^4 c^6 + 8 a^10 b^6 c^6 -
30 a^8 b^8 c^6 + 8 a^6 b^10 c^6 + 20 a^4 b^12 c^6 -
24 a^2 b^14 c^6 + 11 b^16 c^6 - 3 a^14 c^8 + 13 a^12 b^2 c^8 -
29 a^10 b^4 c^8 + 19 a^8 b^6 c^8 + 19 a^6 b^8 c^8 -
29 a^4 b^10 c^8 + 13 a^2 b^12 c^8 - 3 b^14 c^8 - 35 a^12 c^10 -
2 a^10 b^2 c^10 - 7 a^8 b^4 c^10 - 56 a^6 b^6 c^10 -
7 a^4 b^8 c^10 - 2 a^2 b^10 c^10 - 35 b^12 c^10 + 77 a^10 c^12 +
67 a^8 b^2 c^12 + 120 a^6 b^4 c^12 + 120 a^4 b^6 c^12 +
67 a^2 b^8 c^12 + 77 b^10 c^12 - 99 a^8 c^14 - 172 a^6 b^2 c^14 -
230 a^4 b^4 c^14 - 172 a^2 b^6 c^14 - 99 b^8 c^14 + 94 a^6 c^16 +
194 a^4 b^2 c^16 + 194 a^2 b^4 c^16 + 94 b^6 c^16 - 62 a^4 c^18 -
108 a^2 b^2 c^18 - 62 b^4 c^18 + 24 a^2 c^20 + 24 b^2 c^20 - 4 c^22}

Francisco Javier García Capitán
28 December 2011

## Τρίτη, 27 Δεκεμβρίου 2011

### Circumcenters triangle perspective with ABC

Let ABC be a triangle, P a point, A1B1C1 the circumcevian triangle of P and A2B2C2 the antipodal triangle of A1B1C1 (ie A2,B2,C2 are the antipodes of A1,B1,C1 in the circumcircle).

Denote:

Ab:= the orthogonal projection of A2 in BB1
Ac:= the orthogonal projection of A2 in CC1

O1: = the circumcenter of the triangle A2AbAc

Similarly O2 and O3.

For P = H:
ABC, O1O2O3 are perspective.

Perspector?

Generalization?

APH, 27 December 2011

----------------------------------

For P=H, the perspector is X68.

The locus is (line at infinity) + (circumcircle) + (a quintic through H and O)

For P=O the perspector is O.

quintic:

-2 a^6 b^2 c^4 x^4 y + 2 a^2 b^6 c^4 x^4 y + 4 a^4 b^2 c^6 x^4 y -
2 a^2 b^2 c^8 x^4 y - 2 a^8 c^4 x^3 y^2 - 2 a^6 b^2 c^4 x^3 y^2 +
2 a^2 b^6 c^4 x^3 y^2 + 2 b^8 c^4 x^3 y^2 + 7 a^6 c^6 x^3 y^2 +
5 a^4 b^2 c^6 x^3 y^2 - 3 a^2 b^4 c^6 x^3 y^2 - 5 b^6 c^6 x^3 y^2 -
9 a^4 c^8 x^3 y^2 - 4 a^2 b^2 c^8 x^3 y^2 + 3 b^4 c^8 x^3 y^2 +
5 a^2 c^10 x^3 y^2 + b^2 c^10 x^3 y^2 - c^12 x^3 y^2 -
2 a^8 c^4 x^2 y^3 - 2 a^6 b^2 c^4 x^2 y^3 + 2 a^2 b^6 c^4 x^2 y^3 +
2 b^8 c^4 x^2 y^3 + 5 a^6 c^6 x^2 y^3 + 3 a^4 b^2 c^6 x^2 y^3 -
5 a^2 b^4 c^6 x^2 y^3 - 7 b^6 c^6 x^2 y^3 - 3 a^4 c^8 x^2 y^3 +
4 a^2 b^2 c^8 x^2 y^3 + 9 b^4 c^8 x^2 y^3 - a^2 c^10 x^2 y^3 -
5 b^2 c^10 x^2 y^3 + c^12 x^2 y^3 - 2 a^6 b^2 c^4 x y^4 +
2 a^2 b^6 c^4 x y^4 - 4 a^2 b^4 c^6 x y^4 + 2 a^2 b^2 c^8 x y^4 +
2 a^6 b^4 c^2 x^4 z - 4 a^4 b^6 c^2 x^4 z + 2 a^2 b^8 c^2 x^4 z -
2 a^2 b^4 c^6 x^4 z + a^6 b^4 c^2 x^3 y z + a^4 b^6 c^2 x^3 y z -
5 a^2 b^8 c^2 x^3 y z + 3 b^10 c^2 x^3 y z - a^6 b^2 c^4 x^3 y z +
7 a^2 b^6 c^4 x^3 y z - 6 b^8 c^4 x^3 y z - a^4 b^2 c^6 x^3 y z -
7 a^2 b^4 c^6 x^3 y z + 5 a^2 b^2 c^8 x^3 y z + 6 b^4 c^8 x^3 y z -
3 b^2 c^10 x^3 y z - 2 a^10 c^2 x^2 y^2 z +
2 a^8 b^2 c^2 x^2 y^2 z + 4 a^6 b^4 c^2 x^2 y^2 z -
4 a^4 b^6 c^2 x^2 y^2 z - 2 a^2 b^8 c^2 x^2 y^2 z +
2 b^10 c^2 x^2 y^2 z + 5 a^8 c^4 x^2 y^2 z -
6 a^6 b^2 c^4 x^2 y^2 z + 6 a^2 b^6 c^4 x^2 y^2 z -
5 b^8 c^4 x^2 y^2 z - 3 a^6 c^6 x^2 y^2 z +
5 a^4 b^2 c^6 x^2 y^2 z - 5 a^2 b^4 c^6 x^2 y^2 z +
3 b^6 c^6 x^2 y^2 z - a^4 c^8 x^2 y^2 z + b^4 c^8 x^2 y^2 z +
a^2 c^10 x^2 y^2 z - b^2 c^10 x^2 y^2 z - 3 a^10 c^2 x y^3 z +
5 a^8 b^2 c^2 x y^3 z - a^6 b^4 c^2 x y^3 z - a^4 b^6 c^2 x y^3 z +
6 a^8 c^4 x y^3 z - 7 a^6 b^2 c^4 x y^3 z + a^2 b^6 c^4 x y^3 z +
7 a^4 b^2 c^6 x y^3 z + a^2 b^4 c^6 x y^3 z - 6 a^4 c^8 x y^3 z -
5 a^2 b^2 c^8 x y^3 z + 3 a^2 c^10 x y^3 z - 2 a^8 b^2 c^2 y^4 z +
4 a^6 b^4 c^2 y^4 z - 2 a^4 b^6 c^2 y^4 z + 2 a^4 b^2 c^6 y^4 z +
2 a^8 b^4 x^3 z^2 - 7 a^6 b^6 x^3 z^2 + 9 a^4 b^8 x^3 z^2 -
5 a^2 b^10 x^3 z^2 + b^12 x^3 z^2 + 2 a^6 b^4 c^2 x^3 z^2 -
5 a^4 b^6 c^2 x^3 z^2 + 4 a^2 b^8 c^2 x^3 z^2 - b^10 c^2 x^3 z^2 +
3 a^2 b^6 c^4 x^3 z^2 - 3 b^8 c^4 x^3 z^2 - 2 a^2 b^4 c^6 x^3 z^2 +
5 b^6 c^6 x^3 z^2 - 2 b^4 c^8 x^3 z^2 + 2 a^10 b^2 x^2 y z^2 -
5 a^8 b^4 x^2 y z^2 + 3 a^6 b^6 x^2 y z^2 + a^4 b^8 x^2 y z^2 -
a^2 b^10 x^2 y z^2 - 2 a^8 b^2 c^2 x^2 y z^2 +
6 a^6 b^4 c^2 x^2 y z^2 - 5 a^4 b^6 c^2 x^2 y z^2 +
b^10 c^2 x^2 y z^2 - 4 a^6 b^2 c^4 x^2 y z^2 +
5 a^2 b^6 c^4 x^2 y z^2 - b^8 c^4 x^2 y z^2 +
4 a^4 b^2 c^6 x^2 y z^2 - 6 a^2 b^4 c^6 x^2 y z^2 -
3 b^6 c^6 x^2 y z^2 + 2 a^2 b^2 c^8 x^2 y z^2 +
5 b^4 c^8 x^2 y z^2 - 2 b^2 c^10 x^2 y z^2 + a^10 b^2 x y^2 z^2 -
a^8 b^4 x y^2 z^2 - 3 a^6 b^6 x y^2 z^2 + 5 a^4 b^8 x y^2 z^2 -
2 a^2 b^10 x y^2 z^2 - a^10 c^2 x y^2 z^2 +
5 a^6 b^4 c^2 x y^2 z^2 - 6 a^4 b^6 c^2 x y^2 z^2 +
2 a^2 b^8 c^2 x y^2 z^2 + a^8 c^4 x y^2 z^2 -
5 a^6 b^2 c^4 x y^2 z^2 + 4 a^2 b^6 c^4 x y^2 z^2 +
3 a^6 c^6 x y^2 z^2 + 6 a^4 b^2 c^6 x y^2 z^2 -
4 a^2 b^4 c^6 x y^2 z^2 - 5 a^4 c^8 x y^2 z^2 -
2 a^2 b^2 c^8 x y^2 z^2 + 2 a^2 c^10 x y^2 z^2 - a^12 y^3 z^2 +
5 a^10 b^2 y^3 z^2 - 9 a^8 b^4 y^3 z^2 + 7 a^6 b^6 y^3 z^2 -
2 a^4 b^8 y^3 z^2 + a^10 c^2 y^3 z^2 - 4 a^8 b^2 c^2 y^3 z^2 +
5 a^6 b^4 c^2 y^3 z^2 - 2 a^4 b^6 c^2 y^3 z^2 + 3 a^8 c^4 y^3 z^2 -
3 a^6 b^2 c^4 y^3 z^2 - 5 a^6 c^6 y^3 z^2 + 2 a^4 b^2 c^6 y^3 z^2 +
2 a^4 c^8 y^3 z^2 + 2 a^8 b^4 x^2 z^3 - 5 a^6 b^6 x^2 z^3 +
3 a^4 b^8 x^2 z^3 + a^2 b^10 x^2 z^3 - b^12 x^2 z^3 +
2 a^6 b^4 c^2 x^2 z^3 - 3 a^4 b^6 c^2 x^2 z^3 -
4 a^2 b^8 c^2 x^2 z^3 + 5 b^10 c^2 x^2 z^3 + 5 a^2 b^6 c^4 x^2 z^3 -
9 b^8 c^4 x^2 z^3 - 2 a^2 b^4 c^6 x^2 z^3 + 7 b^6 c^6 x^2 z^3 -
2 b^4 c^8 x^2 z^3 + 3 a^10 b^2 x y z^3 - 6 a^8 b^4 x y z^3 +
6 a^4 b^8 x y z^3 - 3 a^2 b^10 x y z^3 - 5 a^8 b^2 c^2 x y z^3 +
7 a^6 b^4 c^2 x y z^3 - 7 a^4 b^6 c^2 x y z^3 +
5 a^2 b^8 c^2 x y z^3 + a^6 b^2 c^4 x y z^3 - a^2 b^6 c^4 x y z^3 +
a^4 b^2 c^6 x y z^3 - a^2 b^4 c^6 x y z^3 + a^12 y^2 z^3 -
a^10 b^2 y^2 z^3 - 3 a^8 b^4 y^2 z^3 + 5 a^6 b^6 y^2 z^3 -
2 a^4 b^8 y^2 z^3 - 5 a^10 c^2 y^2 z^3 + 4 a^8 b^2 c^2 y^2 z^3 +
3 a^6 b^4 c^2 y^2 z^3 - 2 a^4 b^6 c^2 y^2 z^3 + 9 a^8 c^4 y^2 z^3 -
5 a^6 b^2 c^4 y^2 z^3 - 7 a^6 c^6 y^2 z^3 + 2 a^4 b^2 c^6 y^2 z^3 +
2 a^4 c^8 y^2 z^3 + 2 a^6 b^4 c^2 x z^4 - 2 a^2 b^8 c^2 x z^4 +
4 a^2 b^6 c^4 x z^4 - 2 a^2 b^4 c^6 x z^4 + 2 a^8 b^2 c^2 y z^4 -
2 a^4 b^6 c^2 y z^4 - 4 a^6 b^2 c^4 y z^4 + 2 a^4 b^2 c^6 y z^4 = 0.

Francisco Javier García Capitán
28 December 2011

## Πέμπτη, 22 Δεκεμβρίου 2011

### Reflections of cevians

Let ABC be a triangle and A'B'C' the cevian triangle of H.

Denote:

12 := the reflection of AA' in BB'
13 := the reflection of AA' in CC'

23 := the reflection of BB' in CC'
21 := the reflection of BB' in AA'

31 := the reflection of CC' in AA'
32 := the reflection of CC' in BB'

1'2' := the parallel to 12 through B'
1'3' := the parallel to 13 through C'

2'3' := the parallel to 23 through C'
2'1' := the parallel to 21 through A'

3'1' := the parallel to 31 through A'
3'2' := the parallel to 32 through B'

A* := 1'2' /\ 1'3'
B* := 2'3' /\ 2'1'
C* := 3'1' /\ 3'2'

1. The triangles ABC, A*B*C* are perspective (at H)
2. The triangles A'B'C', A*B*C* are perspective (at H)
2. The points A'B'C'A*B*C* are concyclic (on the NPC)

Generalization:

Which is the locus of P such that:

1. The triangles ABC, A*B*C* are perspective ?
2. The triangles A'B'C', A*B*C* are perspective ?
2. The points A'B'C'A*B*C* are conconic ? (when the conic is circle) ?

Variation:

1'2' := the parallel to 12 through C'
1'3' := the parallel to 13 through B'

2'3' := the parallel to 23 through A'
2'1' := the parallel to 21 through C'

3'1' := the parallel to 31 through B'
3'2' := the parallel to 32 through A'

APH 22 December 2011

## Τρίτη, 20 Δεκεμβρίου 2011

### A "COLOR" THEOREM

Let ABC be a triangle, P a point, A'B'C' the cevian triangle of P and X,Y two fixed points.

From A' we draw two RED lines joining it with X,Y
From B' we draw two BLUE lines joining it with X,Y
From C' we draw two GREEN lines joining it with X,Y.

The lines:

Line joining the intersections of RED with BLUE lines other than X,Y
Line joining the intersections of BLUE with GREEN lines other than X,Y
Line joining the intersections of GREEN with RED lines other than X,Y

are concurrent.

The problem without colors is:

Let ABC be a triangle, P a point, A'B'C' the cevian triangle of P
and X,Y two points.

The lines:

(A'X /\ B'Y) \/ (A'Y /\ B'X)

(B'X /\ C'Y) \/ (B'Y /\ C'X)

(C'X /\ A'Y) \/ (C'Y /\ A'X)

are concurrent.

Proof ??

APH 20 December 2011

-------------------------------------

For P=(u:v:w), X=(u1:v1:w1), Y=(u2:v2:w2) the point of concurrence is:

(u2 w1^2 w2 x^3 y^3 + u1 w1 w2^2 x^3 y^3 + u2 v2 w1^2 x^3 y^2 z -
2 u2 v1 w1 w2 x^3 y^2 z - 2 u1 v2 w1 w2 x^3 y^2 z +
u1 v1 w2^2 x^3 y^2 z - u2^2 w1^2 x^2 y^3 z - u1^2 w2^2 x^2 y^3 z -
2 u2 v1 v2 w1 x^3 y z^2 + u1 v2^2 w1 x^3 y z^2 +
u2 v1^2 w2 x^3 y z^2 - 2 u1 v1 v2 w2 x^3 y z^2 +
2 u2^2 v1 w1 x^2 y^2 z^2 + 2 u1^2 v2 w2 x^2 y^2 z^2 -
u1 u2^2 w1 x y^3 z^2 - u1^2 u2 w2 x y^3 z^2 + u2 v1^2 v2 x^3 z^3 +
u1 v1 v2^2 x^3 z^3 - u2^2 v1^2 x^2 y z^3 - u1^2 v2^2 x^2 y z^3 -
u1 u2^2 v1 x y^2 z^3 - u1^2 u2 v2 x y^2 z^3 + 2 u1^2 u2^2 y^3 z^3 :
v2 w1^2 w2 x^3 y^3 + v1 w1 w2^2 x^3 y^3 - v2^2 w1^2 x^3 y^2 z -
v1^2 w2^2 x^3 y^2 z + u2 v2 w1^2 x^2 y^3 z -
2 u2 v1 w1 w2 x^2 y^3 z - 2 u1 v2 w1 w2 x^2 y^3 z +
u1 v1 w2^2 x^2 y^3 z - v1 v2^2 w1 x^3 y z^2 -
v1^2 v2 w2 x^3 y z^2 + 2 u1 v2^2 w1 x^2 y^2 z^2 +
2 u2 v1^2 w2 x^2 y^2 z^2 + u2^2 v1 w1 x y^3 z^2 -
2 u1 u2 v2 w1 x y^3 z^2 - 2 u1 u2 v1 w2 x y^3 z^2 +
u1^2 v2 w2 x y^3 z^2 + 2 v1^2 v2^2 x^3 z^3 - u2 v1^2 v2 x^2 y z^3 -
u1 v1 v2^2 x^2 y z^3 - u2^2 v1^2 x y^2 z^3 - u1^2 v2^2 x y^2 z^3 +
u1 u2^2 v1 y^3 z^3 + u1^2 u2 v2 y^3 z^3 :
2 w1^2 w2^2 x^3 y^3 - v2 w1^2 w2 x^3 y^2 z - v1 w1 w2^2 x^3 y^2 z -
u2 w1^2 w2 x^2 y^3 z - u1 w1 w2^2 x^2 y^3 z - v2^2 w1^2 x^3 y z^2 -
v1^2 w2^2 x^3 y z^2 + 2 u2 v2 w1^2 x^2 y^2 z^2 +
2 u1 v1 w2^2 x^2 y^2 z^2 - u2^2 w1^2 x y^3 z^2 -
u1^2 w2^2 x y^3 z^2 + v1 v2^2 w1 x^3 z^3 + v1^2 v2 w2 x^3 z^3 -
2 u2 v1 v2 w1 x^2 y z^3 + u1 v2^2 w1 x^2 y z^3 +
u2 v1^2 w2 x^2 y z^3 - 2 u1 v1 v2 w2 x^2 y z^3 +
u2^2 v1 w1 x y^2 z^3 - 2 u1 u2 v2 w1 x y^2 z^3 -
2 u1 u2 v1 w2 x y^2 z^3 + u1^2 v2 w2 x y^2 z^3 +
u1 u2^2 w1 y^3 z^3 + u1^2 u2 w2 y^3 z^3).

Francisco Javier García Capitán
21 December 2011

### Parallels to sidelines

Let ABC be a triangle.

Denote:

Ab = AO /\ BH

Ac = AO /\ CH

A' = (Parallel to AC through Ab) /\ (Parallel to AB trough Ac)

Similarly:

Bc = BO /\ CH

Ba = BO /\ AH

B' = (Parallel to BA through Bc) /\ (Parallel to BC through Ba)

and

Ca = CO /\ AH

Cb = CO /\ BH

C' = (Parallel to CB through Ca) /\ (Parallel to CA through Cb)

The triangles ABC, A'B'C' are perspective.

Perspector?

Also, if we replace H with O and O with H, the triangles are perspective:

Generalization:

P,P* = two isogonal conjugate points (instead of O,H)

APH, 20 December 2011

## Δευτέρα, 19 Δεκεμβρίου 2011

### Reflections of AHO,BHO,CHO in the bisectors

Let ABC be a triangle.

Denote:

AabHabOab : = the reflection of AHO in BI

AacHacOac : = the reflection of AHO in CI

Ah : = AabHab /\ AacHac

Ao : = AabOab /\ AacOac

The line AhAo is parallel to BC.

Similarly the lines BhBo, ChCo are parallels to CA, AB, resp.

Which is the homothetic center of the triangles ABC, Triangle bounded by (AhAo, BhBo, ChCo)?

Generalization:
P,P* := two isogonal conjugate points (instead of H,O).
Which is the locus of P such that ABC, Triangle bounded by (ApAp*, BpBp*, CpCp*) are perspective?

APH, 19 December 2011

----------------------------------------

For H and O (or O and H, which is the same), the perspector is X318.

For P and P' = isogonal conjugate of P, the result is true.

The perspector is:

{a (a - b - c) v (b^2 u (u + v) + v (-c^2 u + a^2 (u + v))) w (c^2 u (u + w) + w (-b^2 u + a^2 (u + w))),
b (a - b + c) u (b^2 u (u + v) + v (-c^2 u + a^2 (u + v))) w (-c^2 v (v + w) - w (-a^2 v + b^2 (v + w))), (a + b - c) c u v (c^2 u (u + w) + w (-b^2 u + a^2 (u + w))) (-c^2 v (v + w) - w (-a^2 v + b^2 (v + w)))}

For P = (u:v:w) and P' = (x:y:z) the locus is:

(b c u x + a c v y + a b w z) (-a^2 c^2 u v x y +
b^2 c^2 u v x y + a c^3 u v x y - b c^3 u v x y +
a^2 b^2 u w x z - a b^3 u w x z + b^3 c u w x z -
b^2 c^2 u w x z + a^3 b v w y z - a^2 b^2 v w y z -
a^3 c v w y z + a^2 c^2 v w y z) (a b c u^2 w x^2 y -
b^2 c u^2 w x^2 y - b c^2 u^2 w x^2 y - a^2 c u v w x^2 y +
2 a b c u v w x^2 y - b^2 c u v w x^2 y - b c^2 u v w x^2 y +
c^3 u v w x^2 y - a^2 c v^2 w x^2 y + a b c v^2 w x^2 y +
a b c u w^2 x^2 y - b c^2 u w^2 x^2 y + a b c v w^2 x^2 y +
a b c u^2 w x y^2 - b^2 c u^2 w x y^2 - a^2 c u v w x y^2 +
2 a b c u v w x y^2 - b^2 c u v w x y^2 - a c^2 u v w x y^2 +
c^3 u v w x y^2 - a^2 c v^2 w x y^2 + a b c v^2 w x y^2 -
a c^2 v^2 w x y^2 + a b c u w^2 x y^2 + a b c v w^2 x y^2 -
a c^2 v w^2 x y^2 + a b c u^2 v x^2 z - b^2 c u^2 v x^2 z -
b c^2 u^2 v x^2 z + a b c u v^2 x^2 z - b^2 c u v^2 x^2 z -
a^2 b u v w x^2 z + b^3 u v w x^2 z + 2 a b c u v w x^2 z -
b^2 c u v w x^2 z - b c^2 u v w x^2 z + a b c v^2 w x^2 z -
a^2 b v w^2 x^2 z + a b c v w^2 x^2 z - a^2 c u^2 v x y z +
2 a b c u^2 v x y z - b^2 c u^2 v x y z - b c^2 u^2 v x y z +
c^3 u^2 v x y z - a^2 c u v^2 x y z + 2 a b c u v^2 x y z -
b^2 c u v^2 x y z - a c^2 u v^2 x y z + c^3 u v^2 x y z -
a^2 b u^2 w x y z + b^3 u^2 w x y z + 2 a b c u^2 w x y z -
b^2 c u^2 w x y z - b c^2 u^2 w x y z + 2 a^3 u v w x y z -
2 a^2 b u v w x y z - 2 a b^2 u v w x y z + 2 b^3 u v w x y z -
2 a^2 c u v w x y z + 6 a b c u v w x y z - 2 b^2 c u v w x y z -
2 a c^2 u v w x y z - 2 b c^2 u v w x y z + 2 c^3 u v w x y z +
a^3 v^2 w x y z - a b^2 v^2 w x y z - a^2 c v^2 w x y z +
2 a b c v^2 w x y z - a c^2 v^2 w x y z - a^2 b u w^2 x y z -
a b^2 u w^2 x y z + b^3 u w^2 x y z + 2 a b c u w^2 x y z -
b c^2 u w^2 x y z + a^3 v w^2 x y z - a^2 b v w^2 x y z -
a b^2 v w^2 x y z + 2 a b c v w^2 x y z - a c^2 v w^2 x y z -
a^2 c u^2 v y^2 z + a b c u^2 v y^2 z - a^2 c u v^2 y^2 z +
a b c u v^2 y^2 z - a c^2 u v^2 y^2 z + a b c u^2 w y^2 z +
a^3 u v w y^2 z - a b^2 u v w y^2 z - a^2 c u v w y^2 z +
2 a b c u v w y^2 z - a c^2 u v w y^2 z - a b^2 u w^2 y^2 z +
a b c u w^2 y^2 z + a b c u^2 v x z^2 - b c^2 u^2 v x z^2 +
a b c u v^2 x z^2 - a^2 b u v w x z^2 - a b^2 u v w x z^2 +
b^3 u v w x z^2 + 2 a b c u v w x z^2 - b c^2 u v w x z^2 -
a b^2 v^2 w x z^2 + a b c v^2 w x z^2 - a^2 b v w^2 x z^2 -
a b^2 v w^2 x z^2 + a b c v w^2 x z^2 + a b c u^2 v y z^2 +
a b c u v^2 y z^2 - a c^2 u v^2 y z^2 - a^2 b u^2 w y z^2 +
a b c u^2 w y z^2 + a^3 u v w y z^2 - a^2 b u v w y z^2 -
a b^2 u v w y z^2 + 2 a b c u v w y z^2 - a c^2 u v w y z^2 -
a^2 b u w^2 y z^2 - a b^2 u w^2 y z^2 + a b c u w^2 y z^2) == 0

hence we have a line, a conic and a cubic.

A particular case:

For P = H, the locus for P'=(x:y:z) factors as:

* Trilinear polar of isogonal conjugate of X2183
* Trilinear polar of X63
* Circumcircle
* Circumhyperbola through O and I.

Francisco Javier García Capitán
19 December 2011

## Σάββατο, 17 Δεκεμβρίου 2011

### Reflections of AH,BH,CH

Let ABC be a triangle.

Denote:

La, Lb, Lc := the reflections of AH,BH,CH in AI,BI,CI, resp.

Lab, Lac := the reflections of La in BH, CH, resp.

A* := Lab /\ Lac

Similarly B*, C*

The Triangles ABC, A*B*C* are perspective.

Perspector ?

Generalization:

Special Case: P,P' = isogonal conjugate points.

APH, 17 December 2011

-------------------------------

For H and I, the perspector is X68.

For P and P' = isogonal conjugate of P, for P = H and P' =(u:v:w) and for P=(u:v:w) and P'=I the loci equations are long.

Since it is true for P = H and P' = I, it would be convenient to think in a function f such that f(H)=I and consider P and P'=f(P)

Francisco Javier García Capitán
18 December 2011

## Παρασκευή, 16 Δεκεμβρίου 2011

### Midpoint of ON

Let ABC be a triangle and L1,L2,L3 the external bisectors of the angles BOC,COA,AOB, resp. (they are parallels to BC,CA,AB, resp.)

Ab, Ac := the orthogonal projections of A on L2,L3, resp.

Bc, Ba := the orthogonal projections of B on L3,L1, resp.

Ca, Cb := the orthogonal projections of C on L1,L2, resp.

The Euler lines of AAbAc, BBcaBa, CCaCb are concurrent at
Q = Midpoint of ON.

Generalization:
P instead of O. Locus of P:
1. for ext. or int. bisectors of BPC, CPA, APC
2. for parallels through P to sidelines of ABC ?

APH, 16 December 2011

## Τετάρτη, 14 Δεκεμβρίου 2011

Let ABC be a triangle, P a point, A1B1C1 and A2B2C2 its cevian and cyclocevian triangles, resp.

Denote:

R1 := the radical axis of the NPCs of A1B2C2, A2B1C1

R2 := the radical axis of the NPCs of B1C2A2, B2C1A1

R3 := the radical axis of the NPCs of C1A2B2, C2A1B1

The lines R1,R2,R3 are concurrent.
(and also the radical axis of the NPCs of A1B1C1,A2B2C2)

Point of concurrence?

Variation:

-- A'B'C' = Circumcevian triangle of point P wrt ABC.

Generalization:

-- A1B1C1, A2B2C2 = two triangles inscribed in the same circle.

APH, 14 December 2011

------------------------------------------------

For the triangle case [cyclocevian] this is the point:

{a^2 (-a^4 c^4 u^3 v^3 + b^4 c^4 u^3 v^3 + 2 a^2 c^6 u^3 v^3 -
c^8 u^3 v^3 + a^6 c^2 u^3 v^2 w + 2 a^4 b^2 c^2 u^3 v^2 w -
7 a^2 b^4 c^2 u^3 v^2 w + 4 b^6 c^2 u^3 v^2 w -
3 a^4 c^4 u^3 v^2 w - 6 a^2 b^2 c^4 u^3 v^2 w -
7 b^4 c^4 u^3 v^2 w + 3 a^2 c^6 u^3 v^2 w + 4 b^2 c^6 u^3 v^2 w -
c^8 u^3 v^2 w + a^6 c^2 u^2 v^3 w - 3 a^2 b^4 c^2 u^2 v^3 w +
2 b^6 c^2 u^2 v^3 w - 4 a^4 c^4 u^2 v^3 w -
6 a^2 b^2 c^4 u^2 v^3 w - 6 b^4 c^4 u^2 v^3 w +
5 a^2 c^6 u^2 v^3 w + 6 b^2 c^6 u^2 v^3 w - 2 c^8 u^2 v^3 w +
a^6 b^2 u^3 v w^2 - 3 a^4 b^4 u^3 v w^2 + 3 a^2 b^6 u^3 v w^2 -
b^8 u^3 v w^2 + 2 a^4 b^2 c^2 u^3 v w^2 -
6 a^2 b^4 c^2 u^3 v w^2 + 4 b^6 c^2 u^3 v w^2 -
7 a^2 b^2 c^4 u^3 v w^2 - 7 b^4 c^4 u^3 v w^2 +
4 b^2 c^6 u^3 v w^2 + 2 a^6 b^2 u^2 v^2 w^2 -
6 a^4 b^4 u^2 v^2 w^2 + 6 a^2 b^6 u^2 v^2 w^2 -
2 b^8 u^2 v^2 w^2 + 2 a^6 c^2 u^2 v^2 w^2 -
10 a^2 b^4 c^2 u^2 v^2 w^2 + 8 b^6 c^2 u^2 v^2 w^2 -
6 a^4 c^4 u^2 v^2 w^2 - 10 a^2 b^2 c^4 u^2 v^2 w^2 -
12 b^4 c^4 u^2 v^2 w^2 + 6 a^2 c^6 u^2 v^2 w^2 +
8 b^2 c^6 u^2 v^2 w^2 - 2 c^8 u^2 v^2 w^2 + a^6 b^2 u v^3 w^2 -
3 a^4 b^4 u v^3 w^2 + 3 a^2 b^6 u v^3 w^2 - b^8 u v^3 w^2 +
2 a^6 c^2 u v^3 w^2 - 2 a^4 b^2 c^2 u v^3 w^2 -
4 a^2 b^4 c^2 u v^3 w^2 + 4 b^6 c^2 u v^3 w^2 -
5 a^4 c^4 u v^3 w^2 - 3 a^2 b^2 c^4 u v^3 w^2 -
6 b^4 c^4 u v^3 w^2 + 4 a^2 c^6 u v^3 w^2 + 4 b^2 c^6 u v^3 w^2 -
c^8 u v^3 w^2 - a^4 b^4 u^3 w^3 + 2 a^2 b^6 u^3 w^3 -
b^8 u^3 w^3 + b^4 c^4 u^3 w^3 + a^6 b^2 u^2 v w^3 -
4 a^4 b^4 u^2 v w^3 + 5 a^2 b^6 u^2 v w^3 - 2 b^8 u^2 v w^3 -
6 a^2 b^4 c^2 u^2 v w^3 + 6 b^6 c^2 u^2 v w^3 -
3 a^2 b^2 c^4 u^2 v w^3 - 6 b^4 c^4 u^2 v w^3 +
2 b^2 c^6 u^2 v w^3 + 2 a^6 b^2 u v^2 w^3 - 5 a^4 b^4 u v^2 w^3 +
4 a^2 b^6 u v^2 w^3 - b^8 u v^2 w^3 + a^6 c^2 u v^2 w^3 -
2 a^4 b^2 c^2 u v^2 w^3 - 3 a^2 b^4 c^2 u v^2 w^3 +
4 b^6 c^2 u v^2 w^3 - 3 a^4 c^4 u v^2 w^3 -
4 a^2 b^2 c^4 u v^2 w^3 - 6 b^4 c^4 u v^2 w^3 +
3 a^2 c^6 u v^2 w^3 + 4 b^2 c^6 u v^2 w^3 - c^8 u v^2 w^3 +
a^6 b^2 v^3 w^3 - 2 a^4 b^4 v^3 w^3 + a^2 b^6 v^3 w^3 +
a^6 c^2 v^3 w^3 - a^2 b^4 c^2 v^3 w^3 - 2 a^4 c^4 v^3 w^3 -
a^2 b^2 c^4 v^3 w^3 + a^2 c^6 v^3 w^3),
b^2 (a^4 c^4 u^3 v^3 - b^4 c^4 u^3 v^3 + 2 b^2 c^6 u^3 v^3 -
c^8 u^3 v^3 + 2 a^6 c^2 u^3 v^2 w - 3 a^4 b^2 c^2 u^3 v^2 w +
b^6 c^2 u^3 v^2 w - 6 a^4 c^4 u^3 v^2 w -
6 a^2 b^2 c^4 u^3 v^2 w - 4 b^4 c^4 u^3 v^2 w +
6 a^2 c^6 u^3 v^2 w + 5 b^2 c^6 u^3 v^2 w - 2 c^8 u^3 v^2 w +
4 a^6 c^2 u^2 v^3 w - 7 a^4 b^2 c^2 u^2 v^3 w +
2 a^2 b^4 c^2 u^2 v^3 w + b^6 c^2 u^2 v^3 w -
7 a^4 c^4 u^2 v^3 w - 6 a^2 b^2 c^4 u^2 v^3 w -
3 b^4 c^4 u^2 v^3 w + 4 a^2 c^6 u^2 v^3 w + 3 b^2 c^6 u^2 v^3 w -
c^8 u^2 v^3 w - a^8 u^3 v w^2 + 3 a^6 b^2 u^3 v w^2 -
3 a^4 b^4 u^3 v w^2 + a^2 b^6 u^3 v w^2 + 4 a^6 c^2 u^3 v w^2 -
4 a^4 b^2 c^2 u^3 v w^2 - 2 a^2 b^4 c^2 u^3 v w^2 +
2 b^6 c^2 u^3 v w^2 - 6 a^4 c^4 u^3 v w^2 -
3 a^2 b^2 c^4 u^3 v w^2 - 5 b^4 c^4 u^3 v w^2 +
4 a^2 c^6 u^3 v w^2 + 4 b^2 c^6 u^3 v w^2 - c^8 u^3 v w^2 -
2 a^8 u^2 v^2 w^2 + 6 a^6 b^2 u^2 v^2 w^2 -
6 a^4 b^4 u^2 v^2 w^2 + 2 a^2 b^6 u^2 v^2 w^2 +
8 a^6 c^2 u^2 v^2 w^2 - 10 a^4 b^2 c^2 u^2 v^2 w^2 +
2 b^6 c^2 u^2 v^2 w^2 - 12 a^4 c^4 u^2 v^2 w^2 -
10 a^2 b^2 c^4 u^2 v^2 w^2 - 6 b^4 c^4 u^2 v^2 w^2 +
8 a^2 c^6 u^2 v^2 w^2 + 6 b^2 c^6 u^2 v^2 w^2 -
2 c^8 u^2 v^2 w^2 - a^8 u v^3 w^2 + 3 a^6 b^2 u v^3 w^2 -
3 a^4 b^4 u v^3 w^2 + a^2 b^6 u v^3 w^2 + 4 a^6 c^2 u v^3 w^2 -
6 a^4 b^2 c^2 u v^3 w^2 + 2 a^2 b^4 c^2 u v^3 w^2 -
7 a^4 c^4 u v^3 w^2 - 7 a^2 b^2 c^4 u v^3 w^2 +
4 a^2 c^6 u v^3 w^2 + a^6 b^2 u^3 w^3 - 2 a^4 b^4 u^3 w^3 +
a^2 b^6 u^3 w^3 - a^4 b^2 c^2 u^3 w^3 + b^6 c^2 u^3 w^3 -
a^2 b^2 c^4 u^3 w^3 - 2 b^4 c^4 u^3 w^3 + b^2 c^6 u^3 w^3 -
a^8 u^2 v w^3 + 4 a^6 b^2 u^2 v w^3 - 5 a^4 b^4 u^2 v w^3 +
2 a^2 b^6 u^2 v w^3 + 4 a^6 c^2 u^2 v w^3 -
3 a^4 b^2 c^2 u^2 v w^3 - 2 a^2 b^4 c^2 u^2 v w^3 +
b^6 c^2 u^2 v w^3 - 6 a^4 c^4 u^2 v w^3 -
4 a^2 b^2 c^4 u^2 v w^3 - 3 b^4 c^4 u^2 v w^3 +
4 a^2 c^6 u^2 v w^3 + 3 b^2 c^6 u^2 v w^3 - c^8 u^2 v w^3 -
2 a^8 u v^2 w^3 + 5 a^6 b^2 u v^2 w^3 - 4 a^4 b^4 u v^2 w^3 +
a^2 b^6 u v^2 w^3 + 6 a^6 c^2 u v^2 w^3 -
6 a^4 b^2 c^2 u v^2 w^3 - 6 a^4 c^4 u v^2 w^3 -
3 a^2 b^2 c^4 u v^2 w^3 + 2 a^2 c^6 u v^2 w^3 - a^8 v^3 w^3 +
2 a^6 b^2 v^3 w^3 - a^4 b^4 v^3 w^3 + a^4 c^4 v^3 w^3),
c^2 (a^6 c^2 u^3 v^3 - a^4 b^2 c^2 u^3 v^3 - a^2 b^4 c^2 u^3 v^3 +
b^6 c^2 u^3 v^3 - 2 a^4 c^4 u^3 v^3 - 2 b^4 c^4 u^3 v^3 +
a^2 c^6 u^3 v^3 + b^2 c^6 u^3 v^3 - a^8 u^3 v^2 w +
4 a^6 b^2 u^3 v^2 w - 6 a^4 b^4 u^3 v^2 w + 4 a^2 b^6 u^3 v^2 w -
b^8 u^3 v^2 w + 3 a^6 c^2 u^3 v^2 w - 4 a^4 b^2 c^2 u^3 v^2 w -
3 a^2 b^4 c^2 u^3 v^2 w + 4 b^6 c^2 u^3 v^2 w -
3 a^4 c^4 u^3 v^2 w - 2 a^2 b^2 c^4 u^3 v^2 w -
5 b^4 c^4 u^3 v^2 w + a^2 c^6 u^3 v^2 w + 2 b^2 c^6 u^3 v^2 w -
a^8 u^2 v^3 w + 4 a^6 b^2 u^2 v^3 w - 6 a^4 b^4 u^2 v^3 w +
4 a^2 b^6 u^2 v^3 w - b^8 u^2 v^3 w + 4 a^6 c^2 u^2 v^3 w -
3 a^4 b^2 c^2 u^2 v^3 w - 4 a^2 b^4 c^2 u^2 v^3 w +
3 b^6 c^2 u^2 v^3 w - 5 a^4 c^4 u^2 v^3 w -
2 a^2 b^2 c^4 u^2 v^3 w - 3 b^4 c^4 u^2 v^3 w +
2 a^2 c^6 u^2 v^3 w + b^2 c^6 u^2 v^3 w + 2 a^6 b^2 u^3 v w^2 -
6 a^4 b^4 u^3 v w^2 + 6 a^2 b^6 u^3 v w^2 - 2 b^8 u^3 v w^2 -
3 a^4 b^2 c^2 u^3 v w^2 - 6 a^2 b^4 c^2 u^3 v w^2 +
5 b^6 c^2 u^3 v w^2 - 4 b^4 c^4 u^3 v w^2 + b^2 c^6 u^3 v w^2 -
2 a^8 u^2 v^2 w^2 + 8 a^6 b^2 u^2 v^2 w^2 -
12 a^4 b^4 u^2 v^2 w^2 + 8 a^2 b^6 u^2 v^2 w^2 -
2 b^8 u^2 v^2 w^2 + 6 a^6 c^2 u^2 v^2 w^2 -
10 a^4 b^2 c^2 u^2 v^2 w^2 - 10 a^2 b^4 c^2 u^2 v^2 w^2 +
6 b^6 c^2 u^2 v^2 w^2 - 6 a^4 c^4 u^2 v^2 w^2 -
6 b^4 c^4 u^2 v^2 w^2 + 2 a^2 c^6 u^2 v^2 w^2 +
2 b^2 c^6 u^2 v^2 w^2 - 2 a^8 u v^3 w^2 + 6 a^6 b^2 u v^3 w^2 -
6 a^4 b^4 u v^3 w^2 + 2 a^2 b^6 u v^3 w^2 + 5 a^6 c^2 u v^3 w^2 -
6 a^4 b^2 c^2 u v^3 w^2 - 3 a^2 b^4 c^2 u v^3 w^2 -
4 a^4 c^4 u v^3 w^2 + a^2 c^6 u v^3 w^2 + a^4 b^4 u^3 w^3 -
b^8 u^3 w^3 + 2 b^6 c^2 u^3 w^3 - b^4 c^4 u^3 w^3 +
4 a^6 b^2 u^2 v w^3 - 7 a^4 b^4 u^2 v w^3 + 4 a^2 b^6 u^2 v w^3 -
b^8 u^2 v w^3 - 7 a^4 b^2 c^2 u^2 v w^3 -
6 a^2 b^4 c^2 u^2 v w^3 + 3 b^6 c^2 u^2 v w^3 +
2 a^2 b^2 c^4 u^2 v w^3 - 3 b^4 c^4 u^2 v w^3 +
b^2 c^6 u^2 v w^3 - a^8 u v^2 w^3 + 4 a^6 b^2 u v^2 w^3 -
7 a^4 b^4 u v^2 w^3 + 4 a^2 b^6 u v^2 w^3 + 3 a^6 c^2 u v^2 w^3 -
6 a^4 b^2 c^2 u v^2 w^3 - 7 a^2 b^4 c^2 u v^2 w^3 -
3 a^4 c^4 u v^2 w^3 + 2 a^2 b^2 c^4 u v^2 w^3 +
a^2 c^6 u v^2 w^3 - a^8 v^3 w^3 + a^4 b^4 v^3 w^3 +
2 a^6 c^2 v^3 w^3 - a^4 c^4 v^3 w^3)}

For A'B'C' = circuncevian triangle of P=(u:v:w) with respect to ABC the intersection of the three radical axes is the point:

{-a^4 b^2 c^4 u^4 v^2 + 4 a^2 b^4 c^4 u^4 v^2 - 3 b^6 c^4 u^4 v^2 +
4 a^2 b^2 c^6 u^4 v^2 + 6 b^4 c^6 u^4 v^2 - 3 b^2 c^8 u^4 v^2 -
a^6 c^4 u^3 v^3 + 4 a^4 b^2 c^4 u^3 v^3 - a^2 b^4 c^4 u^3 v^3 -
2 b^6 c^4 u^3 v^3 + 4 a^4 c^6 u^3 v^3 + 6 a^2 b^2 c^6 u^3 v^3 +
6 b^4 c^6 u^3 v^3 - 5 a^2 c^8 u^3 v^3 - 6 b^2 c^8 u^3 v^3 +
2 c^10 u^3 v^3 + 2 a^4 b^2 c^4 u^2 v^4 - 2 a^2 b^4 c^4 u^2 v^4 +
2 a^4 c^6 u^2 v^4 + 4 a^2 b^2 c^6 u^2 v^4 - 2 a^2 c^8 u^2 v^4 +
2 a^6 b^2 c^2 u^4 v w - 7 a^4 b^4 c^2 u^4 v w +
8 a^2 b^6 c^2 u^4 v w - 3 b^8 c^2 u^4 v w - 7 a^4 b^2 c^4 u^4 v w +
3 b^6 c^4 u^4 v w + 8 a^2 b^2 c^6 u^4 v w + 3 b^4 c^6 u^4 v w -
3 b^2 c^8 u^4 v w + 2 a^8 c^2 u^3 v^2 w - 7 a^6 b^2 c^2 u^3 v^2 w +
8 a^4 b^4 c^2 u^3 v^2 w - 3 a^2 b^6 c^2 u^3 v^2 w -
8 a^6 c^4 u^3 v^2 w + 4 a^4 b^2 c^4 u^3 v^2 w +
2 a^2 b^4 c^4 u^3 v^2 w - 2 b^6 c^4 u^3 v^2 w +
12 a^4 c^6 u^3 v^2 w + 9 a^2 b^2 c^6 u^3 v^2 w +
6 b^4 c^6 u^3 v^2 w - 8 a^2 c^8 u^3 v^2 w - 6 b^2 c^8 u^3 v^2 w +
2 c^10 u^3 v^2 w + a^4 b^4 c^2 u^2 v^3 w - 2 a^2 b^6 c^2 u^2 v^3 w +
b^8 c^2 u^2 v^3 w + 6 a^4 b^2 c^4 u^2 v^3 w +
2 a^2 b^4 c^4 u^2 v^3 w - 4 b^6 c^4 u^2 v^3 w + a^4 c^6 u^2 v^3 w +
2 a^2 b^2 c^6 u^2 v^3 w + 6 b^4 c^6 u^2 v^3 w -
2 a^2 c^8 u^2 v^3 w - 4 b^2 c^8 u^2 v^3 w + c^10 u^2 v^3 w +
a^6 b^2 c^2 u v^4 w - 2 a^4 b^4 c^2 u v^4 w + a^2 b^6 c^2 u v^4 w +
3 a^6 c^4 u v^4 w + 4 a^4 b^2 c^4 u v^4 w - 3 a^2 b^4 c^4 u v^4 w -
2 a^4 c^6 u v^4 w + 3 a^2 b^2 c^6 u v^4 w - a^2 c^8 u v^4 w -
a^4 b^4 c^2 u^4 w^2 + 4 a^2 b^6 c^2 u^4 w^2 - 3 b^8 c^2 u^4 w^2 +
4 a^2 b^4 c^4 u^4 w^2 + 6 b^6 c^4 u^4 w^2 - 3 b^4 c^6 u^4 w^2 +
2 a^8 b^2 u^3 v w^2 - 8 a^6 b^4 u^3 v w^2 + 12 a^4 b^6 u^3 v w^2 -
8 a^2 b^8 u^3 v w^2 + 2 b^10 u^3 v w^2 - 7 a^6 b^2 c^2 u^3 v w^2 +
4 a^4 b^4 c^2 u^3 v w^2 + 9 a^2 b^6 c^2 u^3 v w^2 -
6 b^8 c^2 u^3 v w^2 + 8 a^4 b^2 c^4 u^3 v w^2 +
2 a^2 b^4 c^4 u^3 v w^2 + 6 b^6 c^4 u^3 v w^2 -
3 a^2 b^2 c^6 u^3 v w^2 - 2 b^4 c^6 u^3 v w^2 +
2 a^10 u^2 v^2 w^2 - 7 a^8 b^2 u^2 v^2 w^2 +
8 a^6 b^4 u^2 v^2 w^2 - 2 a^4 b^6 u^2 v^2 w^2 -
2 a^2 b^8 u^2 v^2 w^2 + b^10 u^2 v^2 w^2 - 7 a^8 c^2 u^2 v^2 w^2 +
6 a^4 b^4 c^2 u^2 v^2 w^2 + 4 a^2 b^6 c^2 u^2 v^2 w^2 -
3 b^8 c^2 u^2 v^2 w^2 + 8 a^6 c^4 u^2 v^2 w^2 +
6 a^4 b^2 c^4 u^2 v^2 w^2 - 4 a^2 b^4 c^4 u^2 v^2 w^2 +
2 b^6 c^4 u^2 v^2 w^2 - 2 a^4 c^6 u^2 v^2 w^2 +
4 a^2 b^2 c^6 u^2 v^2 w^2 + 2 b^4 c^6 u^2 v^2 w^2 -
2 a^2 c^8 u^2 v^2 w^2 - 3 b^2 c^8 u^2 v^2 w^2 + c^10 u^2 v^2 w^2 +
a^10 u v^3 w^2 - 4 a^8 b^2 u v^3 w^2 + 6 a^6 b^4 u v^3 w^2 -
4 a^4 b^6 u v^3 w^2 + a^2 b^8 u v^3 w^2 - 4 a^8 c^2 u v^3 w^2 +
a^6 b^2 c^2 u v^3 w^2 + 6 a^4 b^4 c^2 u v^3 w^2 -
3 a^2 b^6 c^2 u v^3 w^2 + 5 a^6 c^4 u v^3 w^2 +
3 a^2 b^4 c^4 u v^3 w^2 - 2 a^4 c^6 u v^3 w^2 -
a^2 b^2 c^6 u v^3 w^2 + a^8 c^2 v^4 w^2 - a^4 b^4 c^2 v^4 w^2 +
2 a^4 b^2 c^4 v^4 w^2 - a^4 c^6 v^4 w^2 - a^6 b^4 u^3 w^3 +
4 a^4 b^6 u^3 w^3 - 5 a^2 b^8 u^3 w^3 + 2 b^10 u^3 w^3 +
4 a^4 b^4 c^2 u^3 w^3 + 6 a^2 b^6 c^2 u^3 w^3 - 6 b^8 c^2 u^3 w^3 -
a^2 b^4 c^4 u^3 w^3 + 6 b^6 c^4 u^3 w^3 - 2 b^4 c^6 u^3 w^3 +
a^4 b^6 u^2 v w^3 - 2 a^2 b^8 u^2 v w^3 + b^10 u^2 v w^3 +
6 a^4 b^4 c^2 u^2 v w^3 + 2 a^2 b^6 c^2 u^2 v w^3 -
4 b^8 c^2 u^2 v w^3 + a^4 b^2 c^4 u^2 v w^3 +
2 a^2 b^4 c^4 u^2 v w^3 + 6 b^6 c^4 u^2 v w^3 -
2 a^2 b^2 c^6 u^2 v w^3 - 4 b^4 c^6 u^2 v w^3 + b^2 c^8 u^2 v w^3 +
a^10 u v^2 w^3 - 4 a^8 b^2 u v^2 w^3 + 5 a^6 b^4 u v^2 w^3 -
2 a^4 b^6 u v^2 w^3 - 4 a^8 c^2 u v^2 w^3 + a^6 b^2 c^2 u v^2 w^3 -
a^2 b^6 c^2 u v^2 w^3 + 6 a^6 c^4 u v^2 w^3 +
6 a^4 b^2 c^4 u v^2 w^3 + 3 a^2 b^4 c^4 u v^2 w^3 -
4 a^4 c^6 u v^2 w^3 - 3 a^2 b^2 c^6 u v^2 w^3 + a^2 c^8 u v^2 w^3 -
a^8 b^2 v^3 w^3 + 2 a^6 b^4 v^3 w^3 - a^4 b^6 v^3 w^3 -
a^8 c^2 v^3 w^3 - 4 a^6 b^2 c^2 v^3 w^3 + a^4 b^4 c^2 v^3 w^3 +
2 a^6 c^4 v^3 w^3 + a^4 b^2 c^4 v^3 w^3 - a^4 c^6 v^3 w^3 +
2 a^4 b^6 u^2 w^4 - 2 a^2 b^8 u^2 w^4 + 2 a^4 b^4 c^2 u^2 w^4 +
4 a^2 b^6 c^2 u^2 w^4 - 2 a^2 b^4 c^4 u^2 w^4 + 3 a^6 b^4 u v w^4 -
2 a^4 b^6 u v w^4 - a^2 b^8 u v w^4 + a^6 b^2 c^2 u v w^4 +
4 a^4 b^4 c^2 u v w^4 + 3 a^2 b^6 c^2 u v w^4 -
2 a^4 b^2 c^4 u v w^4 - 3 a^2 b^4 c^4 u v w^4 +
a^2 b^2 c^6 u v w^4 + a^8 b^2 v^2 w^4 - a^4 b^6 v^2 w^4 +
2 a^4 b^4 c^2 v^2 w^4 -
a^4 b^2 c^4 v^2 w^4, -2 a^4 b^2 c^4 u^4 v^2 +
2 a^2 b^4 c^4 u^4 v^2 + 4 a^2 b^2 c^6 u^4 v^2 + 2 b^4 c^6 u^4 v^2 -
2 b^2 c^8 u^4 v^2 - 2 a^6 c^4 u^3 v^3 - a^4 b^2 c^4 u^3 v^3 +
4 a^2 b^4 c^4 u^3 v^3 - b^6 c^4 u^3 v^3 + 6 a^4 c^6 u^3 v^3 +
6 a^2 b^2 c^6 u^3 v^3 + 4 b^4 c^6 u^3 v^3 - 6 a^2 c^8 u^3 v^3 -
5 b^2 c^8 u^3 v^3 + 2 c^10 u^3 v^3 - 3 a^6 c^4 u^2 v^4 +
4 a^4 b^2 c^4 u^2 v^4 - a^2 b^4 c^4 u^2 v^4 + 6 a^4 c^6 u^2 v^4 +
4 a^2 b^2 c^6 u^2 v^4 - 3 a^2 c^8 u^2 v^4 + a^6 b^2 c^2 u^4 v w -
2 a^4 b^4 c^2 u^4 v w + a^2 b^6 c^2 u^4 v w -
3 a^4 b^2 c^4 u^4 v w + 4 a^2 b^4 c^4 u^4 v w + 3 b^6 c^4 u^4 v w +
3 a^2 b^2 c^6 u^4 v w - 2 b^4 c^6 u^4 v w - b^2 c^8 u^4 v w +
a^8 c^2 u^3 v^2 w - 2 a^6 b^2 c^2 u^3 v^2 w +
a^4 b^4 c^2 u^3 v^2 w - 4 a^6 c^4 u^3 v^2 w +
2 a^4 b^2 c^4 u^3 v^2 w + 6 a^2 b^4 c^4 u^3 v^2 w +
6 a^4 c^6 u^3 v^2 w + 2 a^2 b^2 c^6 u^3 v^2 w + b^4 c^6 u^3 v^2 w -
4 a^2 c^8 u^3 v^2 w - 2 b^2 c^8 u^3 v^2 w + c^10 u^3 v^2 w -
3 a^6 b^2 c^2 u^2 v^3 w + 8 a^4 b^4 c^2 u^2 v^3 w -
7 a^2 b^6 c^2 u^2 v^3 w + 2 b^8 c^2 u^2 v^3 w -
2 a^6 c^4 u^2 v^3 w + 2 a^4 b^2 c^4 u^2 v^3 w +
4 a^2 b^4 c^4 u^2 v^3 w - 8 b^6 c^4 u^2 v^3 w +
6 a^4 c^6 u^2 v^3 w + 9 a^2 b^2 c^6 u^2 v^3 w +
12 b^4 c^6 u^2 v^3 w - 6 a^2 c^8 u^2 v^3 w - 8 b^2 c^8 u^2 v^3 w +
2 c^10 u^2 v^3 w - 3 a^8 c^2 u v^4 w + 8 a^6 b^2 c^2 u v^4 w -
7 a^4 b^4 c^2 u v^4 w + 2 a^2 b^6 c^2 u v^4 w + 3 a^6 c^4 u v^4 w -
7 a^2 b^4 c^4 u v^4 w + 3 a^4 c^6 u v^4 w + 8 a^2 b^2 c^6 u v^4 w -
3 a^2 c^8 u v^4 w - a^4 b^4 c^2 u^4 w^2 + b^8 c^2 u^4 w^2 +
2 a^2 b^4 c^4 u^4 w^2 - b^4 c^6 u^4 w^2 + a^8 b^2 u^3 v w^2 -
4 a^6 b^4 u^3 v w^2 + 6 a^4 b^6 u^3 v w^2 - 4 a^2 b^8 u^3 v w^2 +
b^10 u^3 v w^2 - 3 a^6 b^2 c^2 u^3 v w^2 + 6 a^4 b^4 c^2 u^3 v w^2 +
a^2 b^6 c^2 u^3 v w^2 - 4 b^8 c^2 u^3 v w^2 +
3 a^4 b^2 c^4 u^3 v w^2 + 5 b^6 c^4 u^3 v w^2 -
a^2 b^2 c^6 u^3 v w^2 - 2 b^4 c^6 u^3 v w^2 + a^10 u^2 v^2 w^2 -
2 a^8 b^2 u^2 v^2 w^2 - 2 a^6 b^4 u^2 v^2 w^2 +
8 a^4 b^6 u^2 v^2 w^2 - 7 a^2 b^8 u^2 v^2 w^2 +
2 b^10 u^2 v^2 w^2 - 3 a^8 c^2 u^2 v^2 w^2 +
4 a^6 b^2 c^2 u^2 v^2 w^2 + 6 a^4 b^4 c^2 u^2 v^2 w^2 -
7 b^8 c^2 u^2 v^2 w^2 + 2 a^6 c^4 u^2 v^2 w^2 -
4 a^4 b^2 c^4 u^2 v^2 w^2 + 6 a^2 b^4 c^4 u^2 v^2 w^2 +
8 b^6 c^4 u^2 v^2 w^2 + 2 a^4 c^6 u^2 v^2 w^2 +
4 a^2 b^2 c^6 u^2 v^2 w^2 - 2 b^4 c^6 u^2 v^2 w^2 -
3 a^2 c^8 u^2 v^2 w^2 - 2 b^2 c^8 u^2 v^2 w^2 + c^10 u^2 v^2 w^2 +
2 a^10 u v^3 w^2 - 8 a^8 b^2 u v^3 w^2 + 12 a^6 b^4 u v^3 w^2 -
8 a^4 b^6 u v^3 w^2 + 2 a^2 b^8 u v^3 w^2 - 6 a^8 c^2 u v^3 w^2 +
9 a^6 b^2 c^2 u v^3 w^2 + 4 a^4 b^4 c^2 u v^3 w^2 -
7 a^2 b^6 c^2 u v^3 w^2 + 6 a^6 c^4 u v^3 w^2 +
2 a^4 b^2 c^4 u v^3 w^2 + 8 a^2 b^4 c^4 u v^3 w^2 -
2 a^4 c^6 u v^3 w^2 - 3 a^2 b^2 c^6 u v^3 w^2 - 3 a^8 c^2 v^4 w^2 +
4 a^6 b^2 c^2 v^4 w^2 - a^4 b^4 c^2 v^4 w^2 + 6 a^6 c^4 v^4 w^2 +
4 a^4 b^2 c^4 v^4 w^2 - 3 a^4 c^6 v^4 w^2 - a^6 b^4 u^3 w^3 +
2 a^4 b^6 u^3 w^3 - a^2 b^8 u^3 w^3 + a^4 b^4 c^2 u^3 w^3 -
4 a^2 b^6 c^2 u^3 w^3 - b^8 c^2 u^3 w^3 + a^2 b^4 c^4 u^3 w^3 +
2 b^6 c^4 u^3 w^3 - b^4 c^6 u^3 w^3 - 2 a^6 b^4 u^2 v w^3 +
5 a^4 b^6 u^2 v w^3 - 4 a^2 b^8 u^2 v w^3 + b^10 u^2 v w^3 -
a^6 b^2 c^2 u^2 v w^3 + a^2 b^6 c^2 u^2 v w^3 -
4 b^8 c^2 u^2 v w^3 + 3 a^4 b^2 c^4 u^2 v w^3 +
6 a^2 b^4 c^4 u^2 v w^3 + 6 b^6 c^4 u^2 v w^3 -
3 a^2 b^2 c^6 u^2 v w^3 - 4 b^4 c^6 u^2 v w^3 + b^2 c^8 u^2 v w^3 +
a^10 u v^2 w^3 - 2 a^8 b^2 u v^2 w^3 + a^6 b^4 u v^2 w^3 -
4 a^8 c^2 u v^2 w^3 + 2 a^6 b^2 c^2 u v^2 w^3 +
6 a^4 b^4 c^2 u v^2 w^3 + 6 a^6 c^4 u v^2 w^3 +
2 a^4 b^2 c^4 u v^2 w^3 + a^2 b^4 c^4 u v^2 w^3 -
4 a^4 c^6 u v^2 w^3 - 2 a^2 b^2 c^6 u v^2 w^3 + a^2 c^8 u v^2 w^3 +
2 a^10 v^3 w^3 - 5 a^8 b^2 v^3 w^3 + 4 a^6 b^4 v^3 w^3 -
a^4 b^6 v^3 w^3 - 6 a^8 c^2 v^3 w^3 + 6 a^6 b^2 c^2 v^3 w^3 +
4 a^4 b^4 c^2 v^3 w^3 + 6 a^6 c^4 v^3 w^3 - a^4 b^2 c^4 v^3 w^3 -
2 a^4 c^6 v^3 w^3 - a^6 b^4 u^2 w^4 + a^2 b^8 u^2 w^4 +
2 a^4 b^4 c^2 u^2 w^4 - a^2 b^4 c^4 u^2 w^4 - a^8 b^2 u v w^4 -
2 a^6 b^4 u v w^4 + 3 a^4 b^6 u v w^4 + 3 a^6 b^2 c^2 u v w^4 +
4 a^4 b^4 c^2 u v w^4 + a^2 b^6 c^2 u v w^4 -
3 a^4 b^2 c^4 u v w^4 - 2 a^2 b^4 c^4 u v w^4 +
a^2 b^2 c^6 u v w^4 - 2 a^8 b^2 v^2 w^4 + 2 a^6 b^4 v^2 w^4 +
4 a^6 b^2 c^2 v^2 w^4 + 2 a^4 b^4 c^2 v^2 w^4 -
2 a^4 b^2 c^4 v^2 w^4, -a^4 b^2 c^4 u^4 v^2 +
2 a^2 b^4 c^4 u^4 v^2 - b^6 c^4 u^4 v^2 + b^2 c^8 u^4 v^2 -
a^6 c^4 u^3 v^3 + a^4 b^2 c^4 u^3 v^3 + a^2 b^4 c^4 u^3 v^3 -
b^6 c^4 u^3 v^3 + 2 a^4 c^6 u^3 v^3 - 4 a^2 b^2 c^6 u^3 v^3 +
2 b^4 c^6 u^3 v^3 - a^2 c^8 u^3 v^3 - b^2 c^8 u^3 v^3 -
a^6 c^4 u^2 v^4 + 2 a^4 b^2 c^4 u^2 v^4 - a^2 b^4 c^4 u^2 v^4 +
a^2 c^8 u^2 v^4 + a^6 b^2 c^2 u^4 v w - 3 a^4 b^4 c^2 u^4 v w +
3 a^2 b^6 c^2 u^4 v w - b^8 c^2 u^4 v w - 2 a^4 b^2 c^4 u^4 v w +
4 a^2 b^4 c^4 u^4 v w - 2 b^6 c^4 u^4 v w + a^2 b^2 c^6 u^4 v w +
3 b^4 c^6 u^4 v w + a^8 c^2 u^3 v^2 w - 3 a^6 b^2 c^2 u^3 v^2 w +
3 a^4 b^4 c^2 u^3 v^2 w - a^2 b^6 c^2 u^3 v^2 w -
4 a^6 c^4 u^3 v^2 w + 6 a^4 b^2 c^4 u^3 v^2 w -
2 b^6 c^4 u^3 v^2 w + 6 a^4 c^6 u^3 v^2 w + a^2 b^2 c^6 u^3 v^2 w +
5 b^4 c^6 u^3 v^2 w - 4 a^2 c^8 u^3 v^2 w - 4 b^2 c^8 u^3 v^2 w +
c^10 u^3 v^2 w - a^6 b^2 c^2 u^2 v^3 w + 3 a^4 b^4 c^2 u^2 v^3 w -
3 a^2 b^6 c^2 u^2 v^3 w + b^8 c^2 u^2 v^3 w - 2 a^6 c^4 u^2 v^3 w +
6 a^2 b^4 c^4 u^2 v^3 w - 4 b^6 c^4 u^2 v^3 w +
5 a^4 c^6 u^2 v^3 w + a^2 b^2 c^6 u^2 v^3 w + 6 b^4 c^6 u^2 v^3 w -
4 a^2 c^8 u^2 v^3 w - 4 b^2 c^8 u^2 v^3 w + c^10 u^2 v^3 w -
a^8 c^2 u v^4 w + 3 a^6 b^2 c^2 u v^4 w - 3 a^4 b^4 c^2 u v^4 w +
a^2 b^6 c^2 u v^4 w - 2 a^6 c^4 u v^4 w + 4 a^4 b^2 c^4 u v^4 w -
2 a^2 b^4 c^4 u v^4 w + 3 a^4 c^6 u v^4 w + a^2 b^2 c^6 u v^4 w -
2 a^4 b^4 c^2 u^4 w^2 + 4 a^2 b^6 c^2 u^4 w^2 - 2 b^8 c^2 u^4 w^2 +
2 a^2 b^4 c^4 u^4 w^2 + 2 b^6 c^4 u^4 w^2 + a^8 b^2 u^3 v w^2 -
4 a^6 b^4 u^3 v w^2 + 6 a^4 b^6 u^3 v w^2 - 4 a^2 b^8 u^3 v w^2 +
b^10 u^3 v w^2 - 2 a^6 b^2 c^2 u^3 v w^2 +
2 a^4 b^4 c^2 u^3 v w^2 + 2 a^2 b^6 c^2 u^3 v w^2 -
2 b^8 c^2 u^3 v w^2 + a^4 b^2 c^4 u^3 v w^2 +
6 a^2 b^4 c^4 u^3 v w^2 + b^6 c^4 u^3 v w^2 + a^10 u^2 v^2 w^2 -
3 a^8 b^2 u^2 v^2 w^2 + 2 a^6 b^4 u^2 v^2 w^2 +
2 a^4 b^6 u^2 v^2 w^2 - 3 a^2 b^8 u^2 v^2 w^2 + b^10 u^2 v^2 w^2 -
2 a^8 c^2 u^2 v^2 w^2 + 4 a^6 b^2 c^2 u^2 v^2 w^2 -
4 a^4 b^4 c^2 u^2 v^2 w^2 + 4 a^2 b^6 c^2 u^2 v^2 w^2 -
2 b^8 c^2 u^2 v^2 w^2 - 2 a^6 c^4 u^2 v^2 w^2 +
6 a^4 b^2 c^4 u^2 v^2 w^2 + 6 a^2 b^4 c^4 u^2 v^2 w^2 -
2 b^6 c^4 u^2 v^2 w^2 + 8 a^4 c^6 u^2 v^2 w^2 +
8 b^4 c^6 u^2 v^2 w^2 - 7 a^2 c^8 u^2 v^2 w^2 -
7 b^2 c^8 u^2 v^2 w^2 + 2 c^10 u^2 v^2 w^2 + a^10 u v^3 w^2 -
4 a^8 b^2 u v^3 w^2 + 6 a^6 b^4 u v^3 w^2 - 4 a^4 b^6 u v^3 w^2 +
a^2 b^8 u v^3 w^2 - 2 a^8 c^2 u v^3 w^2 + 2 a^6 b^2 c^2 u v^3 w^2 +
2 a^4 b^4 c^2 u v^3 w^2 - 2 a^2 b^6 c^2 u v^3 w^2 +
a^6 c^4 u v^3 w^2 + 6 a^4 b^2 c^4 u v^3 w^2 +
a^2 b^4 c^4 u v^3 w^2 - 2 a^8 c^2 v^4 w^2 + 4 a^6 b^2 c^2 v^4 w^2 -
2 a^4 b^4 c^2 v^4 w^2 + 2 a^6 c^4 v^4 w^2 + 2 a^4 b^2 c^4 v^4 w^2 -
2 a^6 b^4 u^3 w^3 + 6 a^4 b^6 u^3 w^3 - 6 a^2 b^8 u^3 w^3 +
2 b^10 u^3 w^3 - a^4 b^4 c^2 u^3 w^3 + 6 a^2 b^6 c^2 u^3 w^3 -
5 b^8 c^2 u^3 w^3 + 4 a^2 b^4 c^4 u^3 w^3 + 4 b^6 c^4 u^3 w^3 -
b^4 c^6 u^3 w^3 - 2 a^6 b^4 u^2 v w^3 + 6 a^4 b^6 u^2 v w^3 -
6 a^2 b^8 u^2 v w^3 + 2 b^10 u^2 v w^3 - 3 a^6 b^2 c^2 u^2 v w^3 +
2 a^4 b^4 c^2 u^2 v w^3 + 9 a^2 b^6 c^2 u^2 v w^3 -
8 b^8 c^2 u^2 v w^3 + 8 a^4 b^2 c^4 u^2 v w^3 +
4 a^2 b^4 c^4 u^2 v w^3 + 12 b^6 c^4 u^2 v w^3 -
7 a^2 b^2 c^6 u^2 v w^3 - 8 b^4 c^6 u^2 v w^3 +
2 b^2 c^8 u^2 v w^3 + 2 a^10 u v^2 w^3 - 6 a^8 b^2 u v^2 w^3 +
6 a^6 b^4 u v^2 w^3 - 2 a^4 b^6 u v^2 w^3 - 8 a^8 c^2 u v^2 w^3 +
9 a^6 b^2 c^2 u v^2 w^3 + 2 a^4 b^4 c^2 u v^2 w^3 -
3 a^2 b^6 c^2 u v^2 w^3 + 12 a^6 c^4 u v^2 w^3 +
4 a^4 b^2 c^4 u v^2 w^3 + 8 a^2 b^4 c^4 u v^2 w^3 -
8 a^4 c^6 u v^2 w^3 - 7 a^2 b^2 c^6 u v^2 w^3 +
2 a^2 c^8 u v^2 w^3 + 2 a^10 v^3 w^3 - 6 a^8 b^2 v^3 w^3 +
6 a^6 b^4 v^3 w^3 - 2 a^4 b^6 v^3 w^3 - 5 a^8 c^2 v^3 w^3 +
6 a^6 b^2 c^2 v^3 w^3 - a^4 b^4 c^2 v^3 w^3 + 4 a^6 c^4 v^3 w^3 +
4 a^4 b^2 c^4 v^3 w^3 - a^4 c^6 v^3 w^3 - 3 a^6 b^4 u^2 w^4 +
6 a^4 b^6 u^2 w^4 - 3 a^2 b^8 u^2 w^4 + 4 a^4 b^4 c^2 u^2 w^4 +
4 a^2 b^6 c^2 u^2 w^4 - a^2 b^4 c^4 u^2 w^4 - 3 a^8 b^2 u v w^4 +
3 a^6 b^4 u v w^4 + 3 a^4 b^6 u v w^4 - 3 a^2 b^8 u v w^4 +
8 a^6 b^2 c^2 u v w^4 + 8 a^2 b^6 c^2 u v w^4 -
7 a^4 b^2 c^4 u v w^4 - 7 a^2 b^4 c^4 u v w^4 +
2 a^2 b^2 c^6 u v w^4 - 3 a^8 b^2 v^2 w^4 + 6 a^6 b^4 v^2 w^4 -
3 a^4 b^6 v^2 w^4 + 4 a^6 b^2 c^2 v^2 w^4 + 4 a^4 b^4 c^2 v^2 w^4 -
a^4 b^2 c^4 v^2 w^4}

Francisco Javier García Capitán
15 December 2011

------

Generalization:

Let 123456 be a cyclic hexagon (ie inscribed in a circle).

The radical axes of the NPCs of the pairs of the triangles:

abc, def, where {a,b,c,d,e,f} = {1,2,3,4,5,6} (ie pairs of triangles with no common vertex) are concurrent (at the common midpoint of the distances of the centers of the pairs of the NPCs).

We have 10 pairs of triangles:

(123,456), (124,356), (125,346), (126,345)

(134,256), (135,246), (136,245)

(145,236), (146,235)

(156,234)

The centers of the 10 NPCs lie on a conic centered at the point of concurrence of the radical axes.

APH, 17 December 2011

## Τρίτη, 13 Δεκεμβρίου 2011

### Reflctions of AO,BO,CO in BC,CA,AB

Let ABC be a triangle, La the reflection of AO in BC, Lab,Lac the reflections of La in BO, CO, resp. and A' := Lab /\ Lac. Similarly B',C'.

The triangles ABC, A'B'C' are perspective.

Perspector ?

APH, 13 December 2011

--------------------------------------------

The perspector is X26.

The locus is a 16th curve through O and H, which is a trivial case.
Francisco Javier García Capitán
14 December 2011

## Δευτέρα, 12 Δεκεμβρίου 2011

### Parallel Lines : GENERALIZATION 2

[APH]:
> > Let ABC be a triangle and L a line.

> >
> > L1,L2,L3 := the reflections of AI, BI, CI in L, resp.
> >
> > M1,M2,M3 := the reflections of AH, BH, CH in L1, L2, L3, resp
then M1,M2,M3 are parallel.

[Jean-Pierre Ehrmann]:
If (L,L') is the directed angle (mod Pi) between the lines L & L', we have
(M1,M2)+(HA,HB)=2(L1,L2)=2(BI,AI)=(CB,CA)=(HA,HB) thus (M1,M2)=0

Something curious : it seems that, if we take X(80) (reflection of the incenter in the Feuerbach point) instead of H, the lines M1,M2,M3 concur for every line L. But why?

Hyacinthos #20524

[Francisco Javier]:

[APH]:
>It seems that H and X(80) are points of the locus:
>Let ABC be a triangle P a variable point and L a fixed line.
>Let L1, L2, L3 be the reflections of AI,BI,CI in L and
>M1, M2, M3 the reflections of AP,BP,CP in L1,L2,L3, resp.
>Which is the locus of P such that M1,M2,M3 are concurrent?

For each line L: u x + v y + w z = 0 the locus of P=(x:y:z) is a cubic (see
below).

Both X4 and X80 lie on any of these cubics. From a quick sketch, these are the only points lying on all cubics.

Equation of cubic for L: u x + v y + w z = 0 is as follows:

a^5 u^2 x^2 y - 2 a^3 b^2 u^2 x^2 y + a b^4 u^2 x^2 y +
a^3 b c u^2 x^2 y - a b^3 c u^2 x^2 y + a b c^3 u^2 x^2 y -
a c^4 u^2 x^2 y - a^5 u v x^2 y + a^4 b u v x^2 y +
2 a^3 b^2 u v x^2 y - 2 a^2 b^3 u v x^2 y - a b^4 u v x^2 y +
b^5 u v x^2 y - 2 a^3 b c u v x^2 y + 2 a b^3 c u v x^2 y +
2 a^3 c^2 u v x^2 y + a^2 b c^2 u v x^2 y - 2 a b^2 c^2 u v x^2 y -
b^3 c^2 u v x^2 y - a c^4 u v x^2 y - a^4 b v^2 x^2 y +
2 a^2 b^3 v^2 x^2 y - b^5 v^2 x^2 y + a^3 b c v^2 x^2 y -
a b^3 c v^2 x^2 y + a^2 b c^2 v^2 x^2 y - b^3 c^2 v^2 x^2 y -
a b c^3 v^2 x^2 y - 2 a^3 c^2 u w x^2 y + 2 a b^2 c^2 u w x^2 y +
a^2 c^3 u w x^2 y - 2 a b c^3 u w x^2 y + b^2 c^3 u w x^2 y +
2 a c^4 u w x^2 y - c^5 u w x^2 y - 2 a^2 b c^2 v w x^2 y +
2 b^3 c^2 v w x^2 y + 2 a b c^3 v w x^2 y - 2 b^2 c^3 v w x^2 y -
a^2 c^3 w^2 x^2 y + b^2 c^3 w^2 x^2 y + c^5 w^2 x^2 y +
a^5 u^2 x y^2 - 2 a^3 b^2 u^2 x y^2 + a b^4 u^2 x y^2 +
a^3 b c u^2 x y^2 - a b^3 c u^2 x y^2 + a^3 c^2 u^2 x y^2 -
a b^2 c^2 u^2 x y^2 + a b c^3 u^2 x y^2 - a^5 u v x y^2 +
a^4 b u v x y^2 + 2 a^3 b^2 u v x y^2 - 2 a^2 b^3 u v x y^2 -
a b^4 u v x y^2 + b^5 u v x y^2 - 2 a^3 b c u v x y^2 +
2 a b^3 c u v x y^2 + a^3 c^2 u v x y^2 + 2 a^2 b c^2 u v x y^2 -
a b^2 c^2 u v x y^2 - 2 b^3 c^2 u v x y^2 + b c^4 u v x y^2 -
a^4 b v^2 x y^2 + 2 a^2 b^3 v^2 x y^2 - b^5 v^2 x y^2 +
a^3 b c v^2 x y^2 - a b^3 c v^2 x y^2 - a b c^3 v^2 x y^2 +
b c^4 v^2 x y^2 - 2 a^3 c^2 u w x y^2 + 2 a b^2 c^2 u w x y^2 +
2 a^2 c^3 u w x y^2 - 2 a b c^3 u w x y^2 - 2 a^2 b c^2 v w x y^2 +
2 b^3 c^2 v w x y^2 - a^2 c^3 v w x y^2 + 2 a b c^3 v w x y^2 -
b^2 c^3 v w x y^2 - 2 b c^4 v w x y^2 + c^5 v w x y^2 -
a^2 c^3 w^2 x y^2 + b^2 c^3 w^2 x y^2 - c^5 w^2 x y^2 -
a^5 u^2 x^2 z + a b^4 u^2 x^2 z - a^3 b c u^2 x^2 z -
a b^3 c u^2 x^2 z + 2 a^3 c^2 u^2 x^2 z + a b c^3 u^2 x^2 z -
a c^4 u^2 x^2 z + 2 a^3 b^2 u v x^2 z - a^2 b^3 u v x^2 z -
2 a b^4 u v x^2 z + b^5 u v x^2 z + 2 a b^3 c u v x^2 z -
2 a b^2 c^2 u v x^2 z - b^3 c^2 u v x^2 z + a^2 b^3 v^2 x^2 z -
b^5 v^2 x^2 z - b^3 c^2 v^2 x^2 z + a^5 u w x^2 z -
2 a^3 b^2 u w x^2 z + a b^4 u w x^2 z - a^4 c u w x^2 z +
2 a^3 b c u w x^2 z - a^2 b^2 c u w x^2 z - 2 a^3 c^2 u w x^2 z +
2 a b^2 c^2 u w x^2 z + 2 a^2 c^3 u w x^2 z - 2 a b c^3 u w x^2 z +
b^2 c^3 u w x^2 z + a c^4 u w x^2 z - c^5 u w x^2 z +
2 a^2 b^2 c v w x^2 z - 2 a b^3 c v w x^2 z + 2 b^3 c^2 v w x^2 z -
2 b^2 c^3 v w x^2 z + a^4 c w^2 x^2 z - a^3 b c w^2 x^2 z -
a^2 b^2 c w^2 x^2 z + a b^3 c w^2 x^2 z - 2 a^2 c^3 w^2 x^2 z +
a b c^3 w^2 x^2 z + b^2 c^3 w^2 x^2 z + c^5 w^2 x^2 z -
a^3 b^2 u^2 x y z + a b^4 u^2 x y z - 2 a b^3 c u^2 x y z +
a^3 c^2 u^2 x y z + 2 a b c^3 u^2 x y z - a c^4 u^2 x y z -
a^5 u v x y z + 2 a^4 b u v x y z + 3 a^3 b^2 u v x y z -
3 a^2 b^3 u v x y z - 2 a b^4 u v x y z + b^5 u v x y z -
4 a^3 b c u v x y z + 4 a b^3 c u v x y z + 2 a^3 c^2 u v x y z +
a^2 b c^2 u v x y z - a b^2 c^2 u v x y z - 2 b^3 c^2 u v x y z -
a c^4 u v x y z + b c^4 u v x y z - a^4 b v^2 x y z +
a^2 b^3 v^2 x y z + 2 a^3 b c v^2 x y z - b^3 c^2 v^2 x y z -
2 a b c^3 v^2 x y z + b c^4 v^2 x y z + a^5 u w x y z -
2 a^3 b^2 u w x y z + a b^4 u w x y z - 2 a^4 c u w x y z +
4 a^3 b c u w x y z - a^2 b^2 c u w x y z - b^4 c u w x y z -
3 a^3 c^2 u w x y z + a b^2 c^2 u w x y z + 3 a^2 c^3 u w x y z -
4 a b c^3 u w x y z + 2 b^2 c^3 u w x y z + 2 a c^4 u w x y z -
c^5 u w x y z - a^4 b v w x y z + 2 a^2 b^3 v w x y z -
b^5 v w x y z + a^4 c v w x y z + a^2 b^2 c v w x y z -
4 a b^3 c v w x y z + 2 b^4 c v w x y z - a^2 b c^2 v w x y z +
3 b^3 c^2 v w x y z - 2 a^2 c^3 v w x y z + 4 a b c^3 v w x y z -
3 b^2 c^3 v w x y z - 2 b c^4 v w x y z + c^5 v w x y z +
a^4 c w^2 x y z - 2 a^3 b c w^2 x y z + 2 a b^3 c w^2 x y z -
b^4 c w^2 x y z - a^2 c^3 w^2 x y z + b^2 c^3 w^2 x y z +
a^5 u^2 y^2 z - a^3 b^2 u^2 y^2 z + a^3 c^2 u^2 y^2 z -
a^5 u v y^2 z + 2 a^4 b u v y^2 z + a^3 b^2 u v y^2 z -
2 a^2 b^3 u v y^2 z - 2 a^3 b c u v y^2 z + a^3 c^2 u v y^2 z +
2 a^2 b c^2 u v y^2 z - a^4 b v^2 y^2 z + b^5 v^2 y^2 z +
a^3 b c v^2 y^2 z + a b^3 c v^2 y^2 z - 2 b^3 c^2 v^2 y^2 z -
a b c^3 v^2 y^2 z + b c^4 v^2 y^2 z + 2 a^3 b c u w y^2 z -
2 a^2 b^2 c u w y^2 z - 2 a^3 c^2 u w y^2 z + 2 a^2 c^3 u w y^2 z -
a^4 b v w y^2 z + 2 a^2 b^3 v w y^2 z - b^5 v w y^2 z +
a^2 b^2 c v w y^2 z - 2 a b^3 c v w y^2 z + b^4 c v w y^2 z -
2 a^2 b c^2 v w y^2 z + 2 b^3 c^2 v w y^2 z - a^2 c^3 v w y^2 z +
2 a b c^3 v w y^2 z - 2 b^2 c^3 v w y^2 z - b c^4 v w y^2 z +
c^5 v w y^2 z - a^3 b c w^2 y^2 z + a^2 b^2 c w^2 y^2 z +
a b^3 c w^2 y^2 z - b^4 c w^2 y^2 z - a^2 c^3 w^2 y^2 z -
a b c^3 w^2 y^2 z + 2 b^2 c^3 w^2 y^2 z - c^5 w^2 y^2 z -
a^5 u^2 x z^2 - a^3 b^2 u^2 x z^2 - a^3 b c u^2 x z^2 -
a b^3 c u^2 x z^2 + 2 a^3 c^2 u^2 x z^2 + a b^2 c^2 u^2 x z^2 +
a b c^3 u^2 x z^2 - a c^4 u^2 x z^2 + 2 a^3 b^2 u v x z^2 -
2 a^2 b^3 u v x z^2 + 2 a b^3 c u v x z^2 - 2 a b^2 c^2 u v x z^2 +
a^2 b^3 v^2 x z^2 + b^5 v^2 x z^2 - b^3 c^2 v^2 x z^2 +
a^5 u w x z^2 - a^3 b^2 u w x z^2 - a^4 c u w x z^2 +
2 a^3 b c u w x z^2 - 2 a^2 b^2 c u w x z^2 - b^4 c u w x z^2 -
2 a^3 c^2 u w x z^2 + a b^2 c^2 u w x z^2 + 2 a^2 c^3 u w x z^2 -
2 a b c^3 u w x z^2 + 2 b^2 c^3 u w x z^2 + a c^4 u w x z^2 -
c^5 u w x z^2 + a^2 b^3 v w x z^2 - b^5 v w x z^2 +
2 a^2 b^2 c v w x z^2 - 2 a b^3 c v w x z^2 + 2 b^4 c v w x z^2 +
b^3 c^2 v w x z^2 - 2 b^2 c^3 v w x z^2 + a^4 c w^2 x z^2 -
a^3 b c w^2 x z^2 + a b^3 c w^2 x z^2 - b^4 c w^2 x z^2 -
2 a^2 c^3 w^2 x z^2 + a b c^3 w^2 x z^2 + c^5 w^2 x z^2 -
a^5 u^2 y z^2 - a^3 b^2 u^2 y z^2 + a^3 c^2 u^2 y z^2 +
2 a^3 b^2 u v y z^2 - 2 a^2 b^3 u v y z^2 - 2 a^3 b c u v y z^2 +
2 a^2 b c^2 u v y z^2 + a^2 b^3 v^2 y z^2 + b^5 v^2 y z^2 +
a^3 b c v^2 y z^2 + a b^3 c v^2 y z^2 - a^2 b c^2 v^2 y z^2 -
2 b^3 c^2 v^2 y z^2 - a b c^3 v^2 y z^2 + b c^4 v^2 y z^2 +
a^5 u w y z^2 - a^3 b^2 u w y z^2 - 2 a^4 c u w y z^2 +
2 a^3 b c u w y z^2 - 2 a^2 b^2 c u w y z^2 - a^3 c^2 u w y z^2 +
2 a^2 c^3 u w y z^2 + a^2 b^3 v w y z^2 - b^5 v w y z^2 +
a^4 c v w y z^2 + 2 a^2 b^2 c v w y z^2 - 2 a b^3 c v w y z^2 +
b^4 c v w y z^2 - a^2 b c^2 v w y z^2 + 2 b^3 c^2 v w y z^2 -
2 a^2 c^3 v w y z^2 + 2 a b c^3 v w y z^2 - 2 b^2 c^3 v w y z^2 -
b c^4 v w y z^2 + c^5 v w y z^2 + a^4 c w^2 y z^2 -
a^3 b c w^2 y z^2 + a b^3 c w^2 y z^2 - b^4 c w^2 y z^2 -
a b c^3 w^2 y z^2 + 2 b^2 c^3 w^2 y z^2 - c^5 w^2 y z^2 = 0.

Hyacinthos #20527

Francisco Javier García Capitán
12 December 2011

[Jean-Pierre Ehrmann]:
Suppose that P & P' are antipodes upon a rectangular hyperbola.
Consider a line L and a variable point M upon the hyperbola; L'= reflection of MP wrt L; then the reflection of MQ wrt L' goes through a fixed point when M moves upon the hyperbola (this can be shown with an easy computation. Synthetic proof?). If we take successively for M the infinite points of the asymptots, we get an easy localization of the common point.

From this, it follows that if we consider a triangle ABC, a point P, its antigonal P' and a line L. If L1,L2,L3 are the reflections of AP,BP,CP wrt L, then the reflections of AP' wrt L1, of BP' wrt L2, of CP' wrt L3 are concurrent (and if, instead of ABC, we take any triangle inscribed in the rectangular circumhyperbola going through P, the common point will be the same one)
For instance, if P = I, then P'=X[80]

Hyacinthos 20538

### Parallel Lines : GENERALIZATION 1

Let ABC be a triangle, P a point, A'B'C' the pedal triangle of P and L a line.

Denote (for I, J : the incenters of ABC, A'B'C', resp.):
L1,L2,L3 := the reflections of AI, BI, CI in L, resp.
M1,M2,M3 := the reflections of the bisectors A'J, B'J, C'J of the triangle A'B'C' in L1, L2, L3, resp

Which is the locus of P such that the lines M1,M2,M3 are concurrent?

APH, 12 December 2011

## Κυριακή, 11 Δεκεμβρίου 2011

### Parallel Lines

Let ABC be a triangle and L a line.

Denote:
L1,L2,L3 := the reflections of AI, BI, CI in L, resp.
M1,M2,M3 := the reflections of AH, BH, CH in L1, L2, L3, resp

Point of concurrence of M1,M2,M3 ?

Special Cases: L = OH or OI or OK lines

APH, 11 December 2011

Generalization 1
Generalization 2

********************************************

I think that my calculations are right, no interesting point in any case.

OH:

{-a^14 b^2 + 5 a^12 b^4 - 10 a^10 b^6 + 10 a^8 b^8 - 5 a^6 b^10 +
a^4 b^12 - a^14 c^2 - 2 a^12 b^2 c^2 + 4 a^10 b^4 c^2 +
6 a^8 b^6 c^2 - 8 a^6 b^8 c^2 - a^4 b^10 c^2 + a^2 b^12 c^2 +
b^14 c^2 + 5 a^12 c^4 + 4 a^10 b^2 c^4 - 24 a^8 b^4 c^4 +
12 a^6 b^6 c^4 + 12 a^4 b^8 c^4 - 3 a^2 b^10 c^4 - 6 b^12 c^4 -
10 a^10 c^6 + 6 a^8 b^2 c^6 + 12 a^6 b^4 c^6 - 24 a^4 b^6 c^6 +
2 a^2 b^8 c^6 + 15 b^10 c^6 + 10 a^8 c^8 - 8 a^6 b^2 c^8 +
12 a^4 b^4 c^8 + 2 a^2 b^6 c^8 - 20 b^8 c^8 - 5 a^6 c^10 -
a^4 b^2 c^10 - 3 a^2 b^4 c^10 + 15 b^6 c^10 + a^4 c^12 +
a^2 b^2 c^12 - 6 b^4 c^12 + b^2 c^14,
a^12 b^4 - 5 a^10 b^6 + 10 a^8 b^8 - 10 a^6 b^10 + 5 a^4 b^12 -
a^2 b^14 + a^14 c^2 + a^12 b^2 c^2 - a^10 b^4 c^2 - 8 a^8 b^6 c^2 +
6 a^6 b^8 c^2 + 4 a^4 b^10 c^2 - 2 a^2 b^12 c^2 - b^14 c^2 -
6 a^12 c^4 - 3 a^10 b^2 c^4 + 12 a^8 b^4 c^4 + 12 a^6 b^6 c^4 -
24 a^4 b^8 c^4 + 4 a^2 b^10 c^4 + 5 b^12 c^4 + 15 a^10 c^6 +
2 a^8 b^2 c^6 - 24 a^6 b^4 c^6 + 12 a^4 b^6 c^6 + 6 a^2 b^8 c^6 -
10 b^10 c^6 - 20 a^8 c^8 + 2 a^6 b^2 c^8 + 12 a^4 b^4 c^8 -
8 a^2 b^6 c^8 + 10 b^8 c^8 + 15 a^6 c^10 - 3 a^4 b^2 c^10 -
a^2 b^4 c^10 - 5 b^6 c^10 - 6 a^4 c^12 + a^2 b^2 c^12 + b^4 c^12 +
a^2 c^14,
a^14 b^2 - 6 a^12 b^4 + 15 a^10 b^6 - 20 a^8 b^8 + 15 a^6 b^10 -
6 a^4 b^12 + a^2 b^14 + a^12 b^2 c^2 - 3 a^10 b^4 c^2 +
2 a^8 b^6 c^2 + 2 a^6 b^8 c^2 - 3 a^4 b^10 c^2 + a^2 b^12 c^2 +
a^12 c^4 - a^10 b^2 c^4 + 12 a^8 b^4 c^4 - 24 a^6 b^6 c^4 +
12 a^4 b^8 c^4 - a^2 b^10 c^4 + b^12 c^4 - 5 a^10 c^6 -
8 a^8 b^2 c^6 + 12 a^6 b^4 c^6 + 12 a^4 b^6 c^6 - 8 a^2 b^8 c^6 -
5 b^10 c^6 + 10 a^8 c^8 + 6 a^6 b^2 c^8 - 24 a^4 b^4 c^8 +
6 a^2 b^6 c^8 + 10 b^8 c^8 - 10 a^6 c^10 + 4 a^4 b^2 c^10 +
4 a^2 b^4 c^10 - 10 b^6 c^10 + 5 a^4 c^12 - 2 a^2 b^2 c^12 +
5 b^4 c^12 - a^2 c^14 - b^2 c^14}

OI:

{a^2 (a^10 b^6 - 3 a^8 b^8 + 3 a^6 b^10 - a^4 b^12 - a^8 b^6 c^2 -
a^6 b^8 c^2 + 2 a^4 b^10 c^2 + 4 a^6 b^6 c^4 - 5 a^4 b^8 c^4 +
a^10 c^6 - a^8 b^2 c^6 + 4 a^6 b^4 c^6 + a^2 b^8 c^6 - b^10 c^6 -
3 a^8 c^8 - a^6 b^2 c^8 - 5 a^4 b^4 c^8 + a^2 b^6 c^8 +
2 b^8 c^8 + 3 a^6 c^10 + 2 a^4 b^2 c^10 - b^6 c^10 - a^4 c^12),
b^2 (-a^12 b^4 + 3 a^10 b^6 - 3 a^8 b^8 + a^6 b^10 + 2 a^10 b^4 c^2 -
a^8 b^6 c^2 - a^6 b^8 c^2 - 5 a^8 b^4 c^4 + 4 a^6 b^6 c^4 -
a^10 c^6 + a^8 b^2 c^6 + 4 a^4 b^6 c^6 - a^2 b^8 c^6 + b^10 c^6 +
2 a^8 c^8 + a^6 b^2 c^8 - 5 a^4 b^4 c^8 - a^2 b^6 c^8 -
3 b^8 c^8 - a^6 c^10 + 2 a^2 b^4 c^10 + 3 b^6 c^10 - b^4 c^12),
c^2 (-a^10 b^6 + 2 a^8 b^8 - a^6 b^10 + a^8 b^6 c^2 + a^6 b^8 c^2 -
a^12 c^4 + 2 a^10 b^2 c^4 - 5 a^8 b^4 c^4 - 5 a^4 b^8 c^4 +
2 a^2 b^10 c^4 - b^12 c^4 + 3 a^10 c^6 - a^8 b^2 c^6 +
4 a^6 b^4 c^6 + 4 a^4 b^6 c^6 - a^2 b^8 c^6 + 3 b^10 c^6 -
3 a^8 c^8 - a^6 b^2 c^8 - a^2 b^6 c^8 - 3 b^8 c^8 + a^6 c^10 +
b^6 c^10)}

OK:

{a^2 (a^10 b^6 - 3 a^8 b^8 + 3 a^6 b^10 - a^4 b^12 - a^8 b^6 c^2 -
a^6 b^8 c^2 + 2 a^4 b^10 c^2 + 4 a^6 b^6 c^4 - 5 a^4 b^8 c^4 +
a^10 c^6 - a^8 b^2 c^6 + 4 a^6 b^4 c^6 + a^2 b^8 c^6 - b^10 c^6 -
3 a^8 c^8 - a^6 b^2 c^8 - 5 a^4 b^4 c^8 + a^2 b^6 c^8 +
2 b^8 c^8 + 3 a^6 c^10 + 2 a^4 b^2 c^10 - b^6 c^10 - a^4 c^12),
b^2 (-a^12 b^4 + 3 a^10 b^6 - 3 a^8 b^8 + a^6 b^10 + 2 a^10 b^4 c^2 -
a^8 b^6 c^2 - a^6 b^8 c^2 - 5 a^8 b^4 c^4 + 4 a^6 b^6 c^4 -
a^10 c^6 + a^8 b^2 c^6 + 4 a^4 b^6 c^6 - a^2 b^8 c^6 + b^10 c^6 +
2 a^8 c^8 + a^6 b^2 c^8 - 5 a^4 b^4 c^8 - a^2 b^6 c^8 -
3 b^8 c^8 - a^6 c^10 + 2 a^2 b^4 c^10 + 3 b^6 c^10 - b^4 c^12),
c^2 (-a^10 b^6 + 2 a^8 b^8 - a^6 b^10 + a^8 b^6 c^2 + a^6 b^8 c^2 -
a^12 c^4 + 2 a^10 b^2 c^4 - 5 a^8 b^4 c^4 - 5 a^4 b^8 c^4 +
2 a^2 b^10 c^4 - b^12 c^4 + 3 a^10 c^6 - a^8 b^2 c^6 +
4 a^6 b^4 c^6 + 4 a^4 b^6 c^6 - a^2 b^8 c^6 + 3 b^10 c^6 -
3 a^8 c^8 - a^6 b^2 c^8 - a^2 b^6 c^8 - 3 b^8 c^8 + a^6 c^10 +
b^6 c^10)}

The general point of intersection, for a line L = ux+vy+wz=0 is:

{-a^6 b^2 u^4 + a^4 b^4 u^4 - a^6 c^2 u^4 - 2 a^4 b^2 c^2 u^4 +
a^4 c^4 u^4 + 4 a^6 b^2 u^3 v - 4 a^4 b^4 u^3 v +
4 a^4 b^2 c^2 u^3 v - 6 a^6 b^2 u^2 v^2 + 6 a^4 b^4 u^2 v^2 +
6 a^4 b^2 c^2 u^2 v^2 + 4 a^6 b^2 u v^3 - 4 a^4 b^4 u v^3 -
8 a^4 b^2 c^2 u v^3 - 4 a^2 b^4 c^2 u v^3 + 4 a^2 b^2 c^4 u v^3 -
a^6 b^2 v^4 + a^4 b^4 v^4 + 3 a^4 b^2 c^2 v^4 + a^2 b^4 c^2 v^4 +
b^6 c^2 v^4 - 3 a^2 b^2 c^4 v^4 - 2 b^4 c^4 v^4 + b^2 c^6 v^4 +
4 a^6 c^2 u^3 w + 4 a^4 b^2 c^2 u^3 w - 4 a^4 c^4 u^3 w -
24 a^4 b^2 c^2 u^2 v w + 12 a^4 b^2 c^2 u v^2 w +
12 a^2 b^4 c^2 u v^2 w - 12 a^2 b^2 c^4 u v^2 w -
4 a^4 b^2 c^2 v^3 w - 4 b^6 c^2 v^3 w + 8 a^2 b^2 c^4 v^3 w +
8 b^4 c^4 v^3 w - 4 b^2 c^6 v^3 w - 6 a^6 c^2 u^2 w^2 +
6 a^4 b^2 c^2 u^2 w^2 + 6 a^4 c^4 u^2 w^2 +
12 a^4 b^2 c^2 u v w^2 - 12 a^2 b^4 c^2 u v w^2 +
12 a^2 b^2 c^4 u v w^2 - 6 a^2 b^4 c^2 v^2 w^2 +
6 b^6 c^2 v^2 w^2 - 6 a^2 b^2 c^4 v^2 w^2 - 12 b^4 c^4 v^2 w^2 +
6 b^2 c^6 v^2 w^2 + 4 a^6 c^2 u w^3 - 8 a^4 b^2 c^2 u w^3 +
4 a^2 b^4 c^2 u w^3 - 4 a^4 c^4 u w^3 - 4 a^2 b^2 c^4 u w^3 -
4 a^4 b^2 c^2 v w^3 + 8 a^2 b^4 c^2 v w^3 - 4 b^6 c^2 v w^3 +
8 b^4 c^4 v w^3 - 4 b^2 c^6 v w^3 - a^6 c^2 w^4 +
3 a^4 b^2 c^2 w^4 - 3 a^2 b^4 c^2 w^4 + b^6 c^2 w^4 + a^4 c^4 w^4 +
a^2 b^2 c^4 w^4 - 2 b^4 c^4 w^4 + b^2 c^6 w^4,
a^4 b^4 u^4 - a^2 b^6 u^4 + a^6 c^2 u^4 + a^4 b^2 c^2 u^4 +
3 a^2 b^4 c^2 u^4 - 2 a^4 c^4 u^4 - 3 a^2 b^2 c^4 u^4 +
a^2 c^6 u^4 - 4 a^4 b^4 u^3 v + 4 a^2 b^6 u^3 v -
4 a^4 b^2 c^2 u^3 v - 8 a^2 b^4 c^2 u^3 v + 4 a^2 b^2 c^4 u^3 v +
6 a^4 b^4 u^2 v^2 - 6 a^2 b^6 u^2 v^2 + 6 a^2 b^4 c^2 u^2 v^2 -
4 a^4 b^4 u v^3 + 4 a^2 b^6 u v^3 + 4 a^2 b^4 c^2 u v^3 +
a^4 b^4 v^4 - a^2 b^6 v^4 - 2 a^2 b^4 c^2 v^4 - b^6 c^2 v^4 +
b^4 c^4 v^4 - 4 a^6 c^2 u^3 w - 4 a^2 b^4 c^2 u^3 w +
8 a^4 c^4 u^3 w + 8 a^2 b^2 c^4 u^3 w - 4 a^2 c^6 u^3 w +
12 a^4 b^2 c^2 u^2 v w + 12 a^2 b^4 c^2 u^2 v w -
12 a^2 b^2 c^4 u^2 v w - 24 a^2 b^4 c^2 u v^2 w +
4 a^2 b^4 c^2 v^3 w + 4 b^6 c^2 v^3 w - 4 b^4 c^4 v^3 w +
6 a^6 c^2 u^2 w^2 - 6 a^4 b^2 c^2 u^2 w^2 - 12 a^4 c^4 u^2 w^2 -
6 a^2 b^2 c^4 u^2 w^2 + 6 a^2 c^6 u^2 w^2 -
12 a^4 b^2 c^2 u v w^2 + 12 a^2 b^4 c^2 u v w^2 +
12 a^2 b^2 c^4 u v w^2 + 6 a^2 b^4 c^2 v^2 w^2 -
6 b^6 c^2 v^2 w^2 + 6 b^4 c^4 v^2 w^2 - 4 a^6 c^2 u w^3 +
8 a^4 b^2 c^2 u w^3 - 4 a^2 b^4 c^2 u w^3 + 8 a^4 c^4 u w^3 -
4 a^2 c^6 u w^3 + 4 a^4 b^2 c^2 v w^3 - 8 a^2 b^4 c^2 v w^3 +
4 b^6 c^2 v w^3 - 4 a^2 b^2 c^4 v w^3 - 4 b^4 c^4 v w^3 +
a^6 c^2 w^4 - 3 a^4 b^2 c^2 w^4 + 3 a^2 b^4 c^2 w^4 - b^6 c^2 w^4 -
2 a^4 c^4 w^4 + a^2 b^2 c^4 w^4 + b^4 c^4 w^4 + a^2 c^6 w^4,
a^6 b^2 u^4 - 2 a^4 b^4 u^4 + a^2 b^6 u^4 + a^4 b^2 c^2 u^4 -
3 a^2 b^4 c^2 u^4 + a^4 c^4 u^4 + 3 a^2 b^2 c^4 u^4 - a^2 c^6 u^4 -
4 a^6 b^2 u^3 v + 8 a^4 b^4 u^3 v - 4 a^2 b^6 u^3 v +
8 a^2 b^4 c^2 u^3 v - 4 a^2 b^2 c^4 u^3 v + 6 a^6 b^2 u^2 v^2 -
12 a^4 b^4 u^2 v^2 + 6 a^2 b^6 u^2 v^2 - 6 a^4 b^2 c^2 u^2 v^2 -
6 a^2 b^4 c^2 u^2 v^2 - 4 a^6 b^2 u v^3 + 8 a^4 b^4 u v^3 -
4 a^2 b^6 u v^3 + 8 a^4 b^2 c^2 u v^3 - 4 a^2 b^2 c^4 u v^3 +
a^6 b^2 v^4 - 2 a^4 b^4 v^4 + a^2 b^6 v^4 - 3 a^4 b^2 c^2 v^4 +
a^2 b^4 c^2 v^4 + 3 a^2 b^2 c^4 v^4 + b^4 c^4 v^4 - b^2 c^6 v^4 -
4 a^4 b^2 c^2 u^3 w + 4 a^2 b^4 c^2 u^3 w - 4 a^4 c^4 u^3 w -
8 a^2 b^2 c^4 u^3 w + 4 a^2 c^6 u^3 w + 12 a^4 b^2 c^2 u^2 v w -
12 a^2 b^4 c^2 u^2 v w + 12 a^2 b^2 c^4 u^2 v w -
12 a^4 b^2 c^2 u v^2 w + 12 a^2 b^4 c^2 u v^2 w +
12 a^2 b^2 c^4 u v^2 w + 4 a^4 b^2 c^2 v^3 w -
4 a^2 b^4 c^2 v^3 w - 8 a^2 b^2 c^4 v^3 w - 4 b^4 c^4 v^3 w +
4 b^2 c^6 v^3 w + 6 a^4 c^4 u^2 w^2 + 6 a^2 b^2 c^4 u^2 w^2 -
6 a^2 c^6 u^2 w^2 - 24 a^2 b^2 c^4 u v w^2 +
6 a^2 b^2 c^4 v^2 w^2 + 6 b^4 c^4 v^2 w^2 - 6 b^2 c^6 v^2 w^2 -
4 a^4 c^4 u w^3 + 4 a^2 b^2 c^4 u w^3 + 4 a^2 c^6 u w^3 +
4 a^2 b^2 c^4 v w^3 - 4 b^4 c^4 v w^3 + 4 b^2 c^6 v w^3 +
a^4 c^4 w^4 - 2 a^2 b^2 c^4 w^4 + b^4 c^4 w^4 - a^2 c^6 w^4 -
b^2 c^6 w^4}

Francisco Javier García Capitán
12 December 2011

## Παρασκευή, 9 Δεκεμβρίου 2011

### NINE POINT CIRCLE

Let ABC be a triangle, A'B'C' the orthic triangle and P a point.

Let A*,B*,C* be the orthogonal projections of A,B,C on the line OP, resp.
Let L1,L2,L3 be the reflections of A'A*,B'B*,C'C* in the altitudes AA',BB',CC', resp. and M1,M2,M3 the parallels through A,B,C, to L1,L2,L3, resp.

The lines M1,M2,M3 concur at a point Q on the Nine Point Circle of ABC (Q is the center of the rectangular circumhyperbola which is the isogonal conjugate of the line OP)

APH, 9 December 2011

### LOCUS

Generalization of Hyacinthos Message 10485

Let ABC be a triangle Q1, Q2 two fixed points and P a variable point. Let L1,L2,L3 be the parallels through P to AQ2, BQ2, CQ2, respectively.

Ab := L2 /\ (Parallel to BQ1 through A)
Ac := L3 /\ (Parallel to CQ1 through A)

Similarly:

Bc := L3 /\ (Parallel to CQ1 through B)
Ba := L1 /\ (Parallel to AQ1 through B)

Ca := L1 /\ (Parallel to AQ1 through C)
Cb := L2 /\ (Parallel to BQ1 through C)

Which is the locus of P such that the Euler Lines of AAbAc, BBcBa, CCaCb are concurrent?

APH, 9 December 2011

## Τετάρτη, 7 Δεκεμβρίου 2011

### Pedal Triangle. Locus

Let ABC be a triangle, P = (x:y:z) a point, A'B'C' the pedal triangle of P, A"B"C" the circumcevian triangle of A'B'C' with respect its circumcircle (pedal circle of P), and L a line passing through P.

Let A*,B*,C* be the orthogonal projections of A",B",C" on L, resp. The triangles ABC, A*B*C* are perspective. Which is the locus of the perspectors as L moves around P?

APH, 7 December 2011

********************************************

I find that the locus is a quartic whose isotomic conjugate is a conic (it seems that it is always a ellipse).

I give the (long) equation of the conic for an arbitrary point P=(u:v:w).
(I use (u:v:w) for the given point P, then I can use (x:y:z) for a point of the locus when L varies through P.)

16 a^2 c^6 u^4 v^4 x^2 + 4 a^6 c^2 u^4 v^3 w x^2 -
8 a^4 b^2 c^2 u^4 v^3 w x^2 + 4 a^2 b^4 c^2 u^4 v^3 w x^2 -
24 a^4 c^4 u^4 v^3 w x^2 + 40 a^2 b^2 c^4 u^4 v^3 w x^2 +
20 a^2 c^6 u^4 v^3 w x^2 + 4 a^6 c^2 u^3 v^4 w x^2 -
8 a^4 b^2 c^2 u^3 v^4 w x^2 + 4 a^2 b^4 c^2 u^3 v^4 w x^2 +
24 a^4 c^4 u^3 v^4 w x^2 - 8 a^2 b^2 c^4 u^3 v^4 w x^2 +
4 a^2 c^6 u^3 v^4 w x^2 + 4 a^6 b^2 u^4 v^2 w^2 x^2 -
8 a^4 b^4 u^4 v^2 w^2 x^2 + 4 a^2 b^6 u^4 v^2 w^2 x^2 +
4 a^6 c^2 u^4 v^2 w^2 x^2 - 48 a^4 b^2 c^2 u^4 v^2 w^2 x^2 +
44 a^2 b^4 c^2 u^4 v^2 w^2 x^2 - 8 a^4 c^4 u^4 v^2 w^2 x^2 +
44 a^2 b^2 c^4 u^4 v^2 w^2 x^2 + 4 a^2 c^6 u^4 v^2 w^2 x^2 +
4 a^8 u^3 v^3 w^2 x^2 - 4 a^6 b^2 u^3 v^3 w^2 x^2 -
4 a^4 b^4 u^3 v^3 w^2 x^2 + 4 a^2 b^6 u^3 v^3 w^2 x^2 -
32 a^6 c^2 u^3 v^3 w^2 x^2 + 32 a^4 b^2 c^2 u^3 v^3 w^2 x^2 +
20 a^4 c^4 u^3 v^3 w^2 x^2 - 12 a^2 b^2 c^4 u^3 v^3 w^2 x^2 +
8 a^2 c^6 u^3 v^3 w^2 x^2 + 4 a^8 u^2 v^4 w^2 x^2 -
8 a^6 b^2 u^2 v^4 w^2 x^2 + 4 a^4 b^4 u^2 v^4 w^2 x^2 +
8 a^6 c^2 u^2 v^4 w^2 x^2 - 8 a^4 b^2 c^2 u^2 v^4 w^2 x^2 +
4 a^4 c^4 u^2 v^4 w^2 x^2 + 4 a^6 b^2 u^4 v w^3 x^2 -
24 a^4 b^4 u^4 v w^3 x^2 + 20 a^2 b^6 u^4 v w^3 x^2 -
8 a^4 b^2 c^2 u^4 v w^3 x^2 + 40 a^2 b^4 c^2 u^4 v w^3 x^2 +
4 a^2 b^2 c^4 u^4 v w^3 x^2 + 4 a^8 u^3 v^2 w^3 x^2 -
32 a^6 b^2 u^3 v^2 w^3 x^2 + 20 a^4 b^4 u^3 v^2 w^3 x^2 +
8 a^2 b^6 u^3 v^2 w^3 x^2 - 4 a^6 c^2 u^3 v^2 w^3 x^2 +
32 a^4 b^2 c^2 u^3 v^2 w^3 x^2 - 12 a^2 b^4 c^2 u^3 v^2 w^3 x^2 -
4 a^4 c^4 u^3 v^2 w^3 x^2 + 4 a^2 c^6 u^3 v^2 w^3 x^2 -
8 a^8 u^2 v^3 w^3 x^2 + 8 a^4 b^4 u^2 v^3 w^3 x^2 -
16 a^4 b^2 c^2 u^2 v^3 w^3 x^2 + 8 a^4 c^4 u^2 v^3 w^3 x^2 +
16 a^2 b^6 u^4 w^4 x^2 + 4 a^6 b^2 u^3 v w^4 x^2 +
24 a^4 b^4 u^3 v w^4 x^2 + 4 a^2 b^6 u^3 v w^4 x^2 -
8 a^4 b^2 c^2 u^3 v w^4 x^2 - 8 a^2 b^4 c^2 u^3 v w^4 x^2 +
4 a^2 b^2 c^4 u^3 v w^4 x^2 + 4 a^8 u^2 v^2 w^4 x^2 +
8 a^6 b^2 u^2 v^2 w^4 x^2 + 4 a^4 b^4 u^2 v^2 w^4 x^2 -
8 a^6 c^2 u^2 v^2 w^4 x^2 - 8 a^4 b^2 c^2 u^2 v^2 w^4 x^2 +
4 a^4 c^4 u^2 v^2 w^4 x^2 + 16 a^2 c^6 u^4 v^4 x y +
16 b^2 c^6 u^4 v^4 x y - 16 c^8 u^4 v^4 x y +
4 a^6 c^2 u^4 v^3 w x y - 4 a^4 b^2 c^2 u^4 v^3 w x y -
4 a^2 b^4 c^2 u^4 v^3 w x y + 4 b^6 c^2 u^4 v^3 w x y -
16 a^4 c^4 u^4 v^3 w x y + 16 a^2 b^2 c^4 u^4 v^3 w x y +
32 b^4 c^4 u^4 v^3 w x y + 20 a^2 c^6 u^4 v^3 w x y -
28 b^2 c^6 u^4 v^3 w x y - 8 c^8 u^4 v^3 w x y +
4 a^6 c^2 u^3 v^4 w x y - 4 a^4 b^2 c^2 u^3 v^4 w x y -
4 a^2 b^4 c^2 u^3 v^4 w x y + 4 b^6 c^2 u^3 v^4 w x y +
32 a^4 c^4 u^3 v^4 w x y + 16 a^2 b^2 c^4 u^3 v^4 w x y -
16 b^4 c^4 u^3 v^4 w x y - 28 a^2 c^6 u^3 v^4 w x y +
20 b^2 c^6 u^3 v^4 w x y - 8 c^8 u^3 v^4 w x y -
a^8 u^4 v^2 w^2 x y + 8 a^6 b^2 u^4 v^2 w^2 x y -
10 a^4 b^4 u^4 v^2 w^2 x y + 3 b^8 u^4 v^2 w^2 x y +
4 a^6 c^2 u^4 v^2 w^2 x y - 24 a^4 b^2 c^2 u^4 v^2 w^2 x y -
12 a^2 b^4 c^2 u^4 v^2 w^2 x y + 32 b^6 c^2 u^4 v^2 w^2 x y -
6 a^4 c^4 u^4 v^2 w^2 x y + 24 a^2 b^2 c^4 u^4 v^2 w^2 x y -
26 b^4 c^4 u^4 v^2 w^2 x y + 4 a^2 c^6 u^4 v^2 w^2 x y -
8 b^2 c^6 u^4 v^2 w^2 x y - c^8 u^4 v^2 w^2 x y +
2 a^8 u^3 v^3 w^2 x y + 8 a^6 b^2 u^3 v^3 w^2 x y -
20 a^4 b^4 u^3 v^3 w^2 x y + 8 a^2 b^6 u^3 v^3 w^2 x y +
2 b^8 u^3 v^3 w^2 x y - 8 a^6 c^2 u^3 v^3 w^2 x y +
8 a^4 b^2 c^2 u^3 v^3 w^2 x y + 8 a^2 b^4 c^2 u^3 v^3 w^2 x y -
8 b^6 c^2 u^3 v^3 w^2 x y - 80 a^2 b^2 c^4 u^3 v^3 w^2 x y +
16 a^2 c^6 u^3 v^3 w^2 x y + 16 b^2 c^6 u^3 v^3 w^2 x y -
10 c^8 u^3 v^3 w^2 x y + 3 a^8 u^2 v^4 w^2 x y -
10 a^4 b^4 u^2 v^4 w^2 x y + 8 a^2 b^6 u^2 v^4 w^2 x y -
b^8 u^2 v^4 w^2 x y + 32 a^6 c^2 u^2 v^4 w^2 x y -
12 a^4 b^2 c^2 u^2 v^4 w^2 x y - 24 a^2 b^4 c^2 u^2 v^4 w^2 x y +
4 b^6 c^2 u^2 v^4 w^2 x y - 26 a^4 c^4 u^2 v^4 w^2 x y +
24 a^2 b^2 c^4 u^2 v^4 w^2 x y - 6 b^4 c^4 u^2 v^4 w^2 x y -
8 a^2 c^6 u^2 v^4 w^2 x y + 4 b^2 c^6 u^2 v^4 w^2 x y -
c^8 u^2 v^4 w^2 x y - 4 a^4 b^4 u^4 v w^3 x y -
8 a^2 b^6 u^4 v w^3 x y + 12 b^8 u^4 v w^3 x y +
8 a^2 b^4 c^2 u^4 v w^3 x y - 8 b^6 c^2 u^4 v w^3 x y -
4 b^4 c^4 u^4 v w^3 x y - 2 a^8 u^3 v^2 w^3 x y +
4 a^6 b^2 u^3 v^2 w^3 x y - 20 a^4 b^4 u^3 v^2 w^3 x y +
20 a^2 b^6 u^3 v^2 w^3 x y - 2 b^8 u^3 v^2 w^3 x y +
8 a^6 c^2 u^3 v^2 w^3 x y - 12 a^4 b^2 c^2 u^3 v^2 w^3 x y -
88 a^2 b^4 c^2 u^3 v^2 w^3 x y - 4 b^6 c^2 u^3 v^2 w^3 x y -
12 a^4 c^4 u^3 v^2 w^3 x y + 12 a^2 b^2 c^4 u^3 v^2 w^3 x y +
12 b^4 c^4 u^3 v^2 w^3 x y + 8 a^2 c^6 u^3 v^2 w^3 x y -
4 b^2 c^6 u^3 v^2 w^3 x y - 2 c^8 u^3 v^2 w^3 x y -
2 a^8 u^2 v^3 w^3 x y + 20 a^6 b^2 u^2 v^3 w^3 x y -
20 a^4 b^4 u^2 v^3 w^3 x y + 4 a^2 b^6 u^2 v^3 w^3 x y -
2 b^8 u^2 v^3 w^3 x y - 4 a^6 c^2 u^2 v^3 w^3 x y -
88 a^4 b^2 c^2 u^2 v^3 w^3 x y - 12 a^2 b^4 c^2 u^2 v^3 w^3 x y +
8 b^6 c^2 u^2 v^3 w^3 x y + 12 a^4 c^4 u^2 v^3 w^3 x y +
12 a^2 b^2 c^4 u^2 v^3 w^3 x y - 12 b^4 c^4 u^2 v^3 w^3 x y -
4 a^2 c^6 u^2 v^3 w^3 x y + 8 b^2 c^6 u^2 v^3 w^3 x y -
2 c^8 u^2 v^3 w^3 x y + 12 a^8 u v^4 w^3 x y -
8 a^6 b^2 u v^4 w^3 x y - 4 a^4 b^4 u v^4 w^3 x y -
8 a^6 c^2 u v^4 w^3 x y + 8 a^4 b^2 c^2 u v^4 w^3 x y -
4 a^4 c^4 u v^4 w^3 x y - 4 a^4 b^4 u^3 v w^4 x y -
24 a^2 b^6 u^3 v w^4 x y - 4 b^8 u^3 v w^4 x y +
8 a^2 b^4 c^2 u^3 v w^4 x y + 8 b^6 c^2 u^3 v w^4 x y -
4 b^4 c^4 u^3 v w^4 x y - a^8 u^2 v^2 w^4 x y -
4 a^6 b^2 u^2 v^2 w^4 x y - 54 a^4 b^4 u^2 v^2 w^4 x y -
4 a^2 b^6 u^2 v^2 w^4 x y - b^8 u^2 v^2 w^4 x y +
4 a^6 c^2 u^2 v^2 w^4 x y + 12 a^4 b^2 c^2 u^2 v^2 w^4 x y +
12 a^2 b^4 c^2 u^2 v^2 w^4 x y + 4 b^6 c^2 u^2 v^2 w^4 x y -
6 a^4 c^4 u^2 v^2 w^4 x y - 12 a^2 b^2 c^4 u^2 v^2 w^4 x y -
6 b^4 c^4 u^2 v^2 w^4 x y + 4 a^2 c^6 u^2 v^2 w^4 x y +
4 b^2 c^6 u^2 v^2 w^4 x y - c^8 u^2 v^2 w^4 x y -
4 a^8 u v^3 w^4 x y - 24 a^6 b^2 u v^3 w^4 x y -
4 a^4 b^4 u v^3 w^4 x y + 8 a^6 c^2 u v^3 w^4 x y +
8 a^4 b^2 c^2 u v^3 w^4 x y - 4 a^4 c^4 u v^3 w^4 x y +
16 b^2 c^6 u^4 v^4 y^2 + 4 a^4 b^2 c^2 u^4 v^3 w y^2 -
8 a^2 b^4 c^2 u^4 v^3 w y^2 + 4 b^6 c^2 u^4 v^3 w y^2 -
8 a^2 b^2 c^4 u^4 v^3 w y^2 + 24 b^4 c^4 u^4 v^3 w y^2 +
4 b^2 c^6 u^4 v^3 w y^2 + 4 a^4 b^2 c^2 u^3 v^4 w y^2 -
8 a^2 b^4 c^2 u^3 v^4 w y^2 + 4 b^6 c^2 u^3 v^4 w y^2 +
40 a^2 b^2 c^4 u^3 v^4 w y^2 - 24 b^4 c^4 u^3 v^4 w y^2 +
20 b^2 c^6 u^3 v^4 w y^2 + 4 a^4 b^4 u^4 v^2 w^2 y^2 -
8 a^2 b^6 u^4 v^2 w^2 y^2 + 4 b^8 u^4 v^2 w^2 y^2 -
8 a^2 b^4 c^2 u^4 v^2 w^2 y^2 + 8 b^6 c^2 u^4 v^2 w^2 y^2 +
4 b^4 c^4 u^4 v^2 w^2 y^2 + 4 a^6 b^2 u^3 v^3 w^2 y^2 -
4 a^4 b^4 u^3 v^3 w^2 y^2 - 4 a^2 b^6 u^3 v^3 w^2 y^2 +
4 b^8 u^3 v^3 w^2 y^2 + 32 a^2 b^4 c^2 u^3 v^3 w^2 y^2 -
32 b^6 c^2 u^3 v^3 w^2 y^2 - 12 a^2 b^2 c^4 u^3 v^3 w^2 y^2 +
20 b^4 c^4 u^3 v^3 w^2 y^2 + 8 b^2 c^6 u^3 v^3 w^2 y^2 +
4 a^6 b^2 u^2 v^4 w^2 y^2 - 8 a^4 b^4 u^2 v^4 w^2 y^2 +
4 a^2 b^6 u^2 v^4 w^2 y^2 + 44 a^4 b^2 c^2 u^2 v^4 w^2 y^2 -
48 a^2 b^4 c^2 u^2 v^4 w^2 y^2 + 4 b^6 c^2 u^2 v^4 w^2 y^2 +
44 a^2 b^2 c^4 u^2 v^4 w^2 y^2 - 8 b^4 c^4 u^2 v^4 w^2 y^2 +
4 b^2 c^6 u^2 v^4 w^2 y^2 + 8 a^4 b^4 u^3 v^2 w^3 y^2 -
8 b^8 u^3 v^2 w^3 y^2 - 16 a^2 b^4 c^2 u^3 v^2 w^3 y^2 +
8 b^4 c^4 u^3 v^2 w^3 y^2 + 8 a^6 b^2 u^2 v^3 w^3 y^2 +
20 a^4 b^4 u^2 v^3 w^3 y^2 - 32 a^2 b^6 u^2 v^3 w^3 y^2 +
4 b^8 u^2 v^3 w^3 y^2 - 12 a^4 b^2 c^2 u^2 v^3 w^3 y^2 +
32 a^2 b^4 c^2 u^2 v^3 w^3 y^2 - 4 b^6 c^2 u^2 v^3 w^3 y^2 -
4 b^4 c^4 u^2 v^3 w^3 y^2 + 4 b^2 c^6 u^2 v^3 w^3 y^2 +
20 a^6 b^2 u v^4 w^3 y^2 - 24 a^4 b^4 u v^4 w^3 y^2 +
4 a^2 b^6 u v^4 w^3 y^2 + 40 a^4 b^2 c^2 u v^4 w^3 y^2 -
8 a^2 b^4 c^2 u v^4 w^3 y^2 + 4 a^2 b^2 c^4 u v^4 w^3 y^2 +
4 a^4 b^4 u^2 v^2 w^4 y^2 + 8 a^2 b^6 u^2 v^2 w^4 y^2 +
4 b^8 u^2 v^2 w^4 y^2 - 8 a^2 b^4 c^2 u^2 v^2 w^4 y^2 -
8 b^6 c^2 u^2 v^2 w^4 y^2 + 4 b^4 c^4 u^2 v^2 w^4 y^2 +
4 a^6 b^2 u v^3 w^4 y^2 + 24 a^4 b^4 u v^3 w^4 y^2 +
4 a^2 b^6 u v^3 w^4 y^2 - 8 a^4 b^2 c^2 u v^3 w^4 y^2 -
8 a^2 b^4 c^2 u v^3 w^4 y^2 + 4 a^2 b^2 c^4 u v^3 w^4 y^2 +
16 a^6 b^2 v^4 w^4 y^2 - 4 a^4 c^4 u^4 v^3 w x z +
8 a^2 b^2 c^4 u^4 v^3 w x z - 4 b^4 c^4 u^4 v^3 w x z -
8 a^2 c^6 u^4 v^3 w x z - 8 b^2 c^6 u^4 v^3 w x z +
12 c^8 u^4 v^3 w x z - 4 a^4 c^4 u^3 v^4 w x z +
8 a^2 b^2 c^4 u^3 v^4 w x z - 4 b^4 c^4 u^3 v^4 w x z -
24 a^2 c^6 u^3 v^4 w x z + 8 b^2 c^6 u^3 v^4 w x z -
4 c^8 u^3 v^4 w x z - a^8 u^4 v^2 w^2 x z +
4 a^6 b^2 u^4 v^2 w^2 x z - 6 a^4 b^4 u^4 v^2 w^2 x z +
4 a^2 b^6 u^4 v^2 w^2 x z - b^8 u^4 v^2 w^2 x z +
8 a^6 c^2 u^4 v^2 w^2 x z - 24 a^4 b^2 c^2 u^4 v^2 w^2 x z +
24 a^2 b^4 c^2 u^4 v^2 w^2 x z - 8 b^6 c^2 u^4 v^2 w^2 x z -
10 a^4 c^4 u^4 v^2 w^2 x z - 12 a^2 b^2 c^4 u^4 v^2 w^2 x z -
26 b^4 c^4 u^4 v^2 w^2 x z + 32 b^2 c^6 u^4 v^2 w^2 x z +
3 c^8 u^4 v^2 w^2 x z - 2 a^8 u^3 v^3 w^2 x z +
8 a^6 b^2 u^3 v^3 w^2 x z - 12 a^4 b^4 u^3 v^3 w^2 x z +
8 a^2 b^6 u^3 v^3 w^2 x z - 2 b^8 u^3 v^3 w^2 x z +
4 a^6 c^2 u^3 v^3 w^2 x z - 12 a^4 b^2 c^2 u^3 v^3 w^2 x z +
12 a^2 b^4 c^2 u^3 v^3 w^2 x z - 4 b^6 c^2 u^3 v^3 w^2 x z -
20 a^4 c^4 u^3 v^3 w^2 x z - 88 a^2 b^2 c^4 u^3 v^3 w^2 x z +
12 b^4 c^4 u^3 v^3 w^2 x z + 20 a^2 c^6 u^3 v^3 w^2 x z -
4 b^2 c^6 u^3 v^3 w^2 x z - 2 c^8 u^3 v^3 w^2 x z -
a^8 u^2 v^4 w^2 x z + 4 a^6 b^2 u^2 v^4 w^2 x z -
6 a^4 b^4 u^2 v^4 w^2 x z + 4 a^2 b^6 u^2 v^4 w^2 x z -
b^8 u^2 v^4 w^2 x z - 4 a^6 c^2 u^2 v^4 w^2 x z +
12 a^4 b^2 c^2 u^2 v^4 w^2 x z - 12 a^2 b^4 c^2 u^2 v^4 w^2 x z +
4 b^6 c^2 u^2 v^4 w^2 x z - 54 a^4 c^4 u^2 v^4 w^2 x z +
12 a^2 b^2 c^4 u^2 v^4 w^2 x z - 6 b^4 c^4 u^2 v^4 w^2 x z -
4 a^2 c^6 u^2 v^4 w^2 x z + 4 b^2 c^6 u^2 v^4 w^2 x z -
c^8 u^2 v^4 w^2 x z + 4 a^6 b^2 u^4 v w^3 x z -
16 a^4 b^4 u^4 v w^3 x z + 20 a^2 b^6 u^4 v w^3 x z -
8 b^8 u^4 v w^3 x z - 4 a^4 b^2 c^2 u^4 v w^3 x z +
16 a^2 b^4 c^2 u^4 v w^3 x z - 28 b^6 c^2 u^4 v w^3 x z -
4 a^2 b^2 c^4 u^4 v w^3 x z + 32 b^4 c^4 u^4 v w^3 x z +
4 b^2 c^6 u^4 v w^3 x z + 2 a^8 u^3 v^2 w^3 x z -
8 a^6 b^2 u^3 v^2 w^3 x z + 16 a^2 b^6 u^3 v^2 w^3 x z -
10 b^8 u^3 v^2 w^3 x z + 8 a^6 c^2 u^3 v^2 w^3 x z +
8 a^4 b^2 c^2 u^3 v^2 w^3 x z - 80 a^2 b^4 c^2 u^3 v^2 w^3 x z +
16 b^6 c^2 u^3 v^2 w^3 x z - 20 a^4 c^4 u^3 v^2 w^3 x z +
8 a^2 b^2 c^4 u^3 v^2 w^3 x z + 8 a^2 c^6 u^3 v^2 w^3 x z -
8 b^2 c^6 u^3 v^2 w^3 x z + 2 c^8 u^3 v^2 w^3 x z -
2 a^8 u^2 v^3 w^3 x z - 4 a^6 b^2 u^2 v^3 w^3 x z +
12 a^4 b^4 u^2 v^3 w^3 x z - 4 a^2 b^6 u^2 v^3 w^3 x z -
2 b^8 u^2 v^3 w^3 x z + 20 a^6 c^2 u^2 v^3 w^3 x z -
88 a^4 b^2 c^2 u^2 v^3 w^3 x z + 12 a^2 b^4 c^2 u^2 v^3 w^3 x z +
8 b^6 c^2 u^2 v^3 w^3 x z - 20 a^4 c^4 u^2 v^3 w^3 x z -
12 a^2 b^2 c^4 u^2 v^3 w^3 x z - 12 b^4 c^4 u^2 v^3 w^3 x z +
4 a^2 c^6 u^2 v^3 w^3 x z + 8 b^2 c^6 u^2 v^3 w^3 x z -
2 c^8 u^2 v^3 w^3 x z - 4 a^8 u v^4 w^3 x z +
8 a^6 b^2 u v^4 w^3 x z - 4 a^4 b^4 u v^4 w^3 x z -
24 a^6 c^2 u v^4 w^3 x z + 8 a^4 b^2 c^2 u v^4 w^3 x z -
4 a^4 c^4 u v^4 w^3 x z + 16 a^2 b^6 u^4 w^4 x z -
16 b^8 u^4 w^4 x z + 16 b^6 c^2 u^4 w^4 x z +
4 a^6 b^2 u^3 v w^4 x z + 32 a^4 b^4 u^3 v w^4 x z -
28 a^2 b^6 u^3 v w^4 x z - 8 b^8 u^3 v w^4 x z -
4 a^4 b^2 c^2 u^3 v w^4 x z + 16 a^2 b^4 c^2 u^3 v w^4 x z +
20 b^6 c^2 u^3 v w^4 x z - 4 a^2 b^2 c^4 u^3 v w^4 x z -
16 b^4 c^4 u^3 v w^4 x z + 4 b^2 c^6 u^3 v w^4 x z +
3 a^8 u^2 v^2 w^4 x z + 32 a^6 b^2 u^2 v^2 w^4 x z -
26 a^4 b^4 u^2 v^2 w^4 x z - 8 a^2 b^6 u^2 v^2 w^4 x z -
b^8 u^2 v^2 w^4 x z - 12 a^4 b^2 c^2 u^2 v^2 w^4 x z +
24 a^2 b^4 c^2 u^2 v^2 w^4 x z + 4 b^6 c^2 u^2 v^2 w^4 x z -
10 a^4 c^4 u^2 v^2 w^4 x z - 24 a^2 b^2 c^4 u^2 v^2 w^4 x z -
6 b^4 c^4 u^2 v^2 w^4 x z + 8 a^2 c^6 u^2 v^2 w^4 x z +
4 b^2 c^6 u^2 v^2 w^4 x z - c^8 u^2 v^2 w^4 x z +
12 a^8 u v^3 w^4 x z - 8 a^6 b^2 u v^3 w^4 x z -
4 a^4 b^4 u v^3 w^4 x z - 8 a^6 c^2 u v^3 w^4 x z +
8 a^4 b^2 c^2 u v^3 w^4 x z - 4 a^4 c^4 u v^3 w^4 x z -
4 a^4 c^4 u^4 v^3 w y z + 8 a^2 b^2 c^4 u^4 v^3 w y z -
4 b^4 c^4 u^4 v^3 w y z + 8 a^2 c^6 u^4 v^3 w y z -
24 b^2 c^6 u^4 v^3 w y z - 4 c^8 u^4 v^3 w y z -
4 a^4 c^4 u^3 v^4 w y z + 8 a^2 b^2 c^4 u^3 v^4 w y z -
4 b^4 c^4 u^3 v^4 w y z - 8 a^2 c^6 u^3 v^4 w y z -
8 b^2 c^6 u^3 v^4 w y z + 12 c^8 u^3 v^4 w y z -
a^8 u^4 v^2 w^2 y z + 4 a^6 b^2 u^4 v^2 w^2 y z -
6 a^4 b^4 u^4 v^2 w^2 y z + 4 a^2 b^6 u^4 v^2 w^2 y z -
b^8 u^4 v^2 w^2 y z + 4 a^6 c^2 u^4 v^2 w^2 y z -
12 a^4 b^2 c^2 u^4 v^2 w^2 y z + 12 a^2 b^4 c^2 u^4 v^2 w^2 y z -
4 b^6 c^2 u^4 v^2 w^2 y z - 6 a^4 c^4 u^4 v^2 w^2 y z +
12 a^2 b^2 c^4 u^4 v^2 w^2 y z - 54 b^4 c^4 u^4 v^2 w^2 y z +
4 a^2 c^6 u^4 v^2 w^2 y z - 4 b^2 c^6 u^4 v^2 w^2 y z -
c^8 u^4 v^2 w^2 y z - 2 a^8 u^3 v^3 w^2 y z +
8 a^6 b^2 u^3 v^3 w^2 y z - 12 a^4 b^4 u^3 v^3 w^2 y z +
8 a^2 b^6 u^3 v^3 w^2 y z - 2 b^8 u^3 v^3 w^2 y z -
4 a^6 c^2 u^3 v^3 w^2 y z + 12 a^4 b^2 c^2 u^3 v^3 w^2 y z -
12 a^2 b^4 c^2 u^3 v^3 w^2 y z + 4 b^6 c^2 u^3 v^3 w^2 y z +
12 a^4 c^4 u^3 v^3 w^2 y z - 88 a^2 b^2 c^4 u^3 v^3 w^2 y z -
20 b^4 c^4 u^3 v^3 w^2 y z - 4 a^2 c^6 u^3 v^3 w^2 y z +
20 b^2 c^6 u^3 v^3 w^2 y z - 2 c^8 u^3 v^3 w^2 y z -
a^8 u^2 v^4 w^2 y z + 4 a^6 b^2 u^2 v^4 w^2 y z -
6 a^4 b^4 u^2 v^4 w^2 y z + 4 a^2 b^6 u^2 v^4 w^2 y z -
b^8 u^2 v^4 w^2 y z - 8 a^6 c^2 u^2 v^4 w^2 y z +
24 a^4 b^2 c^2 u^2 v^4 w^2 y z - 24 a^2 b^4 c^2 u^2 v^4 w^2 y z +
8 b^6 c^2 u^2 v^4 w^2 y z - 26 a^4 c^4 u^2 v^4 w^2 y z -
12 a^2 b^2 c^4 u^2 v^4 w^2 y z - 10 b^4 c^4 u^2 v^4 w^2 y z +
32 a^2 c^6 u^2 v^4 w^2 y z + 3 c^8 u^2 v^4 w^2 y z -
4 a^4 b^4 u^4 v w^3 y z + 8 a^2 b^6 u^4 v w^3 y z -
4 b^8 u^4 v w^3 y z + 8 a^2 b^4 c^2 u^4 v w^3 y z -
24 b^6 c^2 u^4 v w^3 y z - 4 b^4 c^4 u^4 v w^3 y z -
2 a^8 u^3 v^2 w^3 y z - 4 a^6 b^2 u^3 v^2 w^3 y z +
12 a^4 b^4 u^3 v^2 w^3 y z - 4 a^2 b^6 u^3 v^2 w^3 y z -
2 b^8 u^3 v^2 w^3 y z + 8 a^6 c^2 u^3 v^2 w^3 y z +
12 a^4 b^2 c^2 u^3 v^2 w^3 y z - 88 a^2 b^4 c^2 u^3 v^2 w^3 y z +
20 b^6 c^2 u^3 v^2 w^3 y z - 12 a^4 c^4 u^3 v^2 w^3 y z -
12 a^2 b^2 c^4 u^3 v^2 w^3 y z - 20 b^4 c^4 u^3 v^2 w^3 y z +
8 a^2 c^6 u^3 v^2 w^3 y z + 4 b^2 c^6 u^3 v^2 w^3 y z -
2 c^8 u^3 v^2 w^3 y z - 10 a^8 u^2 v^3 w^3 y z +
16 a^6 b^2 u^2 v^3 w^3 y z - 8 a^2 b^6 u^2 v^3 w^3 y z +
2 b^8 u^2 v^3 w^3 y z + 16 a^6 c^2 u^2 v^3 w^3 y z -
80 a^4 b^2 c^2 u^2 v^3 w^3 y z + 8 a^2 b^4 c^2 u^2 v^3 w^3 y z +
8 b^6 c^2 u^2 v^3 w^3 y z + 8 a^2 b^2 c^4 u^2 v^3 w^3 y z -
20 b^4 c^4 u^2 v^3 w^3 y z - 8 a^2 c^6 u^2 v^3 w^3 y z +
8 b^2 c^6 u^2 v^3 w^3 y z + 2 c^8 u^2 v^3 w^3 y z -
8 a^8 u v^4 w^3 y z + 20 a^6 b^2 u v^4 w^3 y z -
16 a^4 b^4 u v^4 w^3 y z + 4 a^2 b^6 u v^4 w^3 y z -
28 a^6 c^2 u v^4 w^3 y z + 16 a^4 b^2 c^2 u v^4 w^3 y z -
4 a^2 b^4 c^2 u v^4 w^3 y z + 32 a^4 c^4 u v^4 w^3 y z -
4 a^2 b^2 c^4 u v^4 w^3 y z + 4 a^2 c^6 u v^4 w^3 y z -
4 a^4 b^4 u^3 v w^4 y z - 8 a^2 b^6 u^3 v w^4 y z +
12 b^8 u^3 v w^4 y z + 8 a^2 b^4 c^2 u^3 v w^4 y z -
8 b^6 c^2 u^3 v w^4 y z - 4 b^4 c^4 u^3 v w^4 y z -
a^8 u^2 v^2 w^4 y z - 8 a^6 b^2 u^2 v^2 w^4 y z -
26 a^4 b^4 u^2 v^2 w^4 y z + 32 a^2 b^6 u^2 v^2 w^4 y z +
3 b^8 u^2 v^2 w^4 y z + 4 a^6 c^2 u^2 v^2 w^4 y z +
24 a^4 b^2 c^2 u^2 v^2 w^4 y z - 12 a^2 b^4 c^2 u^2 v^2 w^4 y z -
6 a^4 c^4 u^2 v^2 w^4 y z - 24 a^2 b^2 c^4 u^2 v^2 w^4 y z -
10 b^4 c^4 u^2 v^2 w^4 y z + 4 a^2 c^6 u^2 v^2 w^4 y z +
8 b^2 c^6 u^2 v^2 w^4 y z - c^8 u^2 v^2 w^4 y z -
8 a^8 u v^3 w^4 y z - 28 a^6 b^2 u v^3 w^4 y z +
32 a^4 b^4 u v^3 w^4 y z + 4 a^2 b^6 u v^3 w^4 y z +
20 a^6 c^2 u v^3 w^4 y z + 16 a^4 b^2 c^2 u v^3 w^4 y z -
4 a^2 b^4 c^2 u v^3 w^4 y z - 16 a^4 c^4 u v^3 w^4 y z -
4 a^2 b^2 c^4 u v^3 w^4 y z + 4 a^2 c^6 u v^3 w^4 y z -
16 a^8 v^4 w^4 y z + 16 a^6 b^2 v^4 w^4 y z +
16 a^6 c^2 v^4 w^4 y z + 4 a^4 c^4 u^4 v^2 w^2 z^2 -
8 a^2 b^2 c^4 u^4 v^2 w^2 z^2 + 4 b^4 c^4 u^4 v^2 w^2 z^2 -
8 a^2 c^6 u^4 v^2 w^2 z^2 + 8 b^2 c^6 u^4 v^2 w^2 z^2 +
4 c^8 u^4 v^2 w^2 z^2 + 8 a^4 c^4 u^3 v^3 w^2 z^2 -
16 a^2 b^2 c^4 u^3 v^3 w^2 z^2 + 8 b^4 c^4 u^3 v^3 w^2 z^2 -
8 c^8 u^3 v^3 w^2 z^2 + 4 a^4 c^4 u^2 v^4 w^2 z^2 -
8 a^2 b^2 c^4 u^2 v^4 w^2 z^2 + 4 b^4 c^4 u^2 v^4 w^2 z^2 +
8 a^2 c^6 u^2 v^4 w^2 z^2 - 8 b^2 c^6 u^2 v^4 w^2 z^2 +
4 c^8 u^2 v^4 w^2 z^2 + 4 a^4 b^2 c^2 u^4 v w^3 z^2 -
8 a^2 b^4 c^2 u^4 v w^3 z^2 + 4 b^6 c^2 u^4 v w^3 z^2 -
8 a^2 b^2 c^4 u^4 v w^3 z^2 + 24 b^4 c^4 u^4 v w^3 z^2 +
4 b^2 c^6 u^4 v w^3 z^2 + 4 a^6 c^2 u^3 v^2 w^3 z^2 -
12 a^2 b^4 c^2 u^3 v^2 w^3 z^2 + 8 b^6 c^2 u^3 v^2 w^3 z^2 -
4 a^4 c^4 u^3 v^2 w^3 z^2 + 32 a^2 b^2 c^4 u^3 v^2 w^3 z^2 +
20 b^4 c^4 u^3 v^2 w^3 z^2 - 4 a^2 c^6 u^3 v^2 w^3 z^2 -
32 b^2 c^6 u^3 v^2 w^3 z^2 + 4 c^8 u^3 v^2 w^3 z^2 +
8 a^6 c^2 u^2 v^3 w^3 z^2 - 12 a^4 b^2 c^2 u^2 v^3 w^3 z^2 +
4 b^6 c^2 u^2 v^3 w^3 z^2 + 20 a^4 c^4 u^2 v^3 w^3 z^2 +
32 a^2 b^2 c^4 u^2 v^3 w^3 z^2 - 4 b^4 c^4 u^2 v^3 w^3 z^2 -
32 a^2 c^6 u^2 v^3 w^3 z^2 - 4 b^2 c^6 u^2 v^3 w^3 z^2 +
4 c^8 u^2 v^3 w^3 z^2 + 4 a^6 c^2 u v^4 w^3 z^2 -
8 a^4 b^2 c^2 u v^4 w^3 z^2 + 4 a^2 b^4 c^2 u v^4 w^3 z^2 +
24 a^4 c^4 u v^4 w^3 z^2 - 8 a^2 b^2 c^4 u v^4 w^3 z^2 +
4 a^2 c^6 u v^4 w^3 z^2 + 16 b^6 c^2 u^4 w^4 z^2 +
4 a^4 b^2 c^2 u^3 v w^4 z^2 + 40 a^2 b^4 c^2 u^3 v w^4 z^2 +
20 b^6 c^2 u^3 v w^4 z^2 - 8 a^2 b^2 c^4 u^3 v w^4 z^2 -
24 b^4 c^4 u^3 v w^4 z^2 + 4 b^2 c^6 u^3 v w^4 z^2 +
4 a^6 c^2 u^2 v^2 w^4 z^2 + 44 a^4 b^2 c^2 u^2 v^2 w^4 z^2 +
44 a^2 b^4 c^2 u^2 v^2 w^4 z^2 + 4 b^6 c^2 u^2 v^2 w^4 z^2 -
8 a^4 c^4 u^2 v^2 w^4 z^2 - 48 a^2 b^2 c^4 u^2 v^2 w^4 z^2 -
8 b^4 c^4 u^2 v^2 w^4 z^2 + 4 a^2 c^6 u^2 v^2 w^4 z^2 +
4 b^2 c^6 u^2 v^2 w^4 z^2 + 20 a^6 c^2 u v^3 w^4 z^2 +
40 a^4 b^2 c^2 u v^3 w^4 z^2 + 4 a^2 b^4 c^2 u v^3 w^4 z^2 -
24 a^4 c^4 u v^3 w^4 z^2 - 8 a^2 b^2 c^4 u v^3 w^4 z^2 +
4 a^2 c^6 u v^3 w^4 z^2 + 16 a^6 c^2 v^4 w^4 z^2 = 0.

[Note: A1B1C1 is A'B'C' and A2B2C2 is A"B"C" in the problem]

Francisco Javier García Capitán
7 December 2011

## Δευτέρα, 5 Δεκεμβρίου 2011

### Reflections of the Circumcircle in the cevians

Let ABC be a triangle and P a point. Let (O1),(O2), (O3) be the reflections of the circumcircle (O) in BC, CA, AB, resp.

Let (O11), (O22), (O33) be the reflections of (O1), (O2), (O3) in AP, BP, CP, resp., L1,L2,L3 the radical axes of [(O22),(O33)], [(O33),(O11)], [(O11), (O22)], resp. and M1,M2,M3 the parallels to L1,L2,L3 through A,B,C resp.

Which is the locus of P such that M1,M2,M3 are concurrent?

APH, 5 December 2011

***********************************

Locus:

(-a - b + c) (a - b + c) (-a + b + c) (a + b + c) (x + y +
z) (-a^4 c^4 x^3 y^2 + a^2 b^2 c^4 x^3 y^2 + 2 a^2 c^6 x^3 y^2 +
b^2 c^6 x^3 y^2 - c^8 x^3 y^2 - a^2 b^2 c^4 x^2 y^3 +
b^4 c^4 x^2 y^3 - a^2 c^6 x^2 y^3 - 2 b^2 c^6 x^2 y^3 +
c^8 x^2 y^3 - a^6 b^2 x^3 y z + 3 a^4 b^4 x^3 y z -
3 a^2 b^6 x^3 y z + b^8 x^3 y z + a^6 c^2 x^3 y z -
b^6 c^2 x^3 y z - 3 a^4 c^4 x^3 y z + 3 a^2 c^6 x^3 y z +
b^2 c^6 x^3 y z - c^8 x^3 y z - a^8 x^2 y^2 z +
2 a^6 b^2 x^2 y^2 z - 2 a^2 b^6 x^2 y^2 z + b^8 x^2 y^2 z +
2 a^6 c^2 x^2 y^2 z - 2 b^6 c^2 x^2 y^2 z - a^4 c^4 x^2 y^2 z +
b^4 c^4 x^2 y^2 z - a^8 x y^3 z + 3 a^6 b^2 x y^3 z -
3 a^4 b^4 x y^3 z + a^2 b^6 x y^3 z + a^6 c^2 x y^3 z -
b^6 c^2 x y^3 z + 3 b^4 c^4 x y^3 z - a^2 c^6 x y^3 z -
3 b^2 c^6 x y^3 z + c^8 x y^3 z + a^4 b^4 x^3 z^2 -
2 a^2 b^6 x^3 z^2 + b^8 x^3 z^2 - a^2 b^4 c^2 x^3 z^2 -
b^6 c^2 x^3 z^2 + a^8 x^2 y z^2 - 2 a^6 b^2 x^2 y z^2 +
a^4 b^4 x^2 y z^2 - 2 a^6 c^2 x^2 y z^2 - b^4 c^4 x^2 y z^2 +
2 a^2 c^6 x^2 y z^2 + 2 b^2 c^6 x^2 y z^2 - c^8 x^2 y z^2 -
a^4 b^4 x y^2 z^2 + 2 a^2 b^6 x y^2 z^2 - b^8 x y^2 z^2 +
2 b^6 c^2 x y^2 z^2 + a^4 c^4 x y^2 z^2 - 2 a^2 c^6 x y^2 z^2 -
2 b^2 c^6 x y^2 z^2 + c^8 x y^2 z^2 - a^8 y^3 z^2 +
2 a^6 b^2 y^3 z^2 - a^4 b^4 y^3 z^2 + a^6 c^2 y^3 z^2 +
a^4 b^2 c^2 y^3 z^2 + a^2 b^6 x^2 z^3 - b^8 x^2 z^3 +
a^2 b^4 c^2 x^2 z^3 + 2 b^6 c^2 x^2 z^3 - b^4 c^4 x^2 z^3 +
a^8 x y z^3 - a^6 b^2 x y z^3 + a^2 b^6 x y z^3 - b^8 x y z^3 -
3 a^6 c^2 x y z^3 + 3 b^6 c^2 x y z^3 + 3 a^4 c^4 x y z^3 -
3 b^4 c^4 x y z^3 - a^2 c^6 x y z^3 + b^2 c^6 x y z^3 +
a^8 y^2 z^3 - a^6 b^2 y^2 z^3 - 2 a^6 c^2 y^2 z^3 -
a^4 b^2 c^2 y^2 z^3 + a^4 c^4 y^2 z^3) (-a^2 c^4 x^4 y^2 +
b^2 c^4 x^4 y^2 + c^6 x^4 y^2 - c^6 x^3 y^3 + a^2 c^4 x^2 y^4 -
b^2 c^4 x^2 y^4 + c^6 x^2 y^4 + a^6 x^4 y z - 2 a^4 b^2 x^4 y z +
a^2 b^4 x^4 y z - 2 a^4 c^2 x^4 y z + 2 b^4 c^2 x^4 y z +
a^2 c^4 x^4 y z + 2 b^2 c^4 x^4 y z + 2 a^6 x^3 y^2 z -
3 a^4 b^2 x^3 y^2 z + b^6 x^3 y^2 z - 4 a^4 c^2 x^3 y^2 z +
4 a^2 b^2 c^2 x^3 y^2 z - 6 b^2 c^4 x^3 y^2 z + 2 c^6 x^3 y^2 z +
a^6 x^2 y^3 z - 3 a^2 b^4 x^2 y^3 z + 2 b^6 x^2 y^3 z +
4 a^2 b^2 c^2 x^2 y^3 z - 4 b^4 c^2 x^2 y^3 z -
6 a^2 c^4 x^2 y^3 z + 2 c^6 x^2 y^3 z + a^4 b^2 x y^4 z -
2 a^2 b^4 x y^4 z + b^6 x y^4 z + 2 a^4 c^2 x y^4 z -
2 b^4 c^2 x y^4 z + 2 a^2 c^4 x y^4 z + b^2 c^4 x y^4 z -
a^2 b^4 x^4 z^2 + b^6 x^4 z^2 + b^4 c^2 x^4 z^2 +
2 a^6 x^3 y z^2 - 4 a^4 b^2 x^3 y z^2 + 2 b^6 x^3 y z^2 -
3 a^4 c^2 x^3 y z^2 + 4 a^2 b^2 c^2 x^3 y z^2 -
6 b^4 c^2 x^3 y z^2 + c^6 x^3 y z^2 + 3 a^6 x^2 y^2 z^2 -
3 a^4 b^2 x^2 y^2 z^2 - 3 a^2 b^4 x^2 y^2 z^2 +
3 b^6 x^2 y^2 z^2 - 3 a^4 c^2 x^2 y^2 z^2 -
6 a^2 b^2 c^2 x^2 y^2 z^2 - 3 b^4 c^2 x^2 y^2 z^2 -
3 a^2 c^4 x^2 y^2 z^2 - 3 b^2 c^4 x^2 y^2 z^2 +
3 c^6 x^2 y^2 z^2 + 2 a^6 x y^3 z^2 - 4 a^2 b^4 x y^3 z^2 +
2 b^6 x y^3 z^2 - 6 a^4 c^2 x y^3 z^2 + 4 a^2 b^2 c^2 x y^3 z^2 -
3 b^4 c^2 x y^3 z^2 + c^6 x y^3 z^2 + a^6 y^4 z^2 -
a^4 b^2 y^4 z^2 + a^4 c^2 y^4 z^2 - b^6 x^3 z^3 + a^6 x^2 y z^3 -
6 a^2 b^4 x^2 y z^3 + 2 b^6 x^2 y z^3 + 4 a^2 b^2 c^2 x^2 y z^3 -
3 a^2 c^4 x^2 y z^3 - 4 b^2 c^4 x^2 y z^3 + 2 c^6 x^2 y z^3 +
2 a^6 x y^2 z^3 - 6 a^4 b^2 x y^2 z^3 + b^6 x y^2 z^3 +
4 a^2 b^2 c^2 x y^2 z^3 - 4 a^2 c^4 x y^2 z^3 -
3 b^2 c^4 x y^2 z^3 + 2 c^6 x y^2 z^3 - a^6 y^3 z^3 +
a^2 b^4 x^2 z^4 + b^6 x^2 z^4 - b^4 c^2 x^2 z^4 +
2 a^4 b^2 x y z^4 + 2 a^2 b^4 x y z^4 + a^4 c^2 x y z^4 +
b^4 c^2 x y z^4 - 2 a^2 c^4 x y z^4 - 2 b^2 c^4 x y z^4 +
c^6 x y z^4 + a^6 y^2 z^4 + a^4 b^2 y^2 z^4 - a^4 c^2 y^2 z^4)

Francisco Javier García Capitán
5 December 2011