## 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