## Παρασκευή, 23 Μαρτίου 2012

### CIRCUMCIRCLE

A circumcircle with thousands of inscribed memories....

## Δευτέρα, 19 Μαρτίου 2012

### Perspective

Let ABC be a triangle, A1B1C1 the orthic triangle, A2B2C2 the circumcevian triangle of O with respect A1B1C1, A3B3C3 the antipodal triangle of A2B2C2 and A'B'C' the triangle bounded by the lines A1A3, B1B3, C1C3, resp..

The triangles A1B1C1 and A'B'C' are perspective.

Perspector?

APH, 19 March 2012

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The perspector is X25.

Francisco Javier García Capitán
19 March 2012

## Παρασκευή, 16 Μαρτίου 2012

### Perspective

Let ABC be a triangle, A'B'C' the cevian triangle of G and A"B"C" the circumcevian triangle of G with respect A'B'C'. The circles with diameters HA",HB",HC" intersect the NPC again at A1,B1,C1, resp.

The circles with diameters HA",HB",HC" intersect the NPC again at A1,B1,C1, resp.

The triangles ABC, A1B1C1 are perspective.

Perspector?

APH, 16 March 2012

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

Not in ETC. It is the isotomic conjugate of

(SA (b^4 + c^4-a^4), SB (a^4 - b^4 + c^4), SC (a^4 + b^4 - c^4))

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

Perspector: X(13854)
Isotomic conjugate: X(34254)
************************************

The locus of P, instead of H, is trilinear polar of X648 (containing H) + NPC + a cubic:

5 a^12 x^3 - 28 a^10 b^2 x^3 + 53 a^8 b^4 x^3 - 40 a^6 b^6 x^3 +
7 a^4 b^8 x^3 + 4 a^2 b^10 x^3 - b^12 x^3 - 28 a^10 c^2 x^3 +
110 a^8 b^2 c^2 x^3 - 120 a^6 b^4 c^2 x^3 + 36 a^4 b^6 c^2 x^3 +
4 a^2 b^8 c^2 x^3 - 2 b^10 c^2 x^3 + 53 a^8 c^4 x^3 -
120 a^6 b^2 c^4 x^3 + 42 a^4 b^4 c^4 x^3 - 8 a^2 b^6 c^4 x^3 +
b^8 c^4 x^3 - 40 a^6 c^6 x^3 + 36 a^4 b^2 c^6 x^3 -
8 a^2 b^4 c^6 x^3 + 4 b^6 c^6 x^3 + 7 a^4 c^8 x^3 +
4 a^2 b^2 c^8 x^3 + b^4 c^8 x^3 + 4 a^2 c^10 x^3 - 2 b^2 c^10 x^3 -
c^12 x^3 + a^12 x^2 y - 4 a^10 b^2 x^2 y - 7 a^8 b^4 x^2 y +
40 a^6 b^6 x^2 y - 53 a^4 b^8 x^2 y + 28 a^2 b^10 x^2 y -
5 b^12 x^2 y - 2 a^10 c^2 x^2 y + 24 a^8 b^2 c^2 x^2 y -
44 a^6 b^4 c^2 x^2 y + 48 a^4 b^6 c^2 x^2 y - 34 a^2 b^8 c^2 x^2 y +
8 b^10 c^2 x^2 y - 17 a^8 c^4 x^2 y + 46 a^4 b^4 c^4 x^2 y +
3 b^8 c^4 x^2 y + 36 a^6 c^6 x^2 y - 24 a^4 b^2 c^6 x^2 y +
4 a^2 b^4 c^6 x^2 y - 8 b^6 c^6 x^2 y - 17 a^4 c^8 x^2 y +
4 a^2 b^2 c^8 x^2 y + b^4 c^8 x^2 y - 2 a^2 c^10 x^2 y +
c^12 x^2 y - 5 a^12 x y^2 + 28 a^10 b^2 x y^2 - 53 a^8 b^4 x y^2 +
40 a^6 b^6 x y^2 - 7 a^4 b^8 x y^2 - 4 a^2 b^10 x y^2 + b^12 x y^2 +
8 a^10 c^2 x y^2 - 34 a^8 b^2 c^2 x y^2 + 48 a^6 b^4 c^2 x y^2 -
44 a^4 b^6 c^2 x y^2 + 24 a^2 b^8 c^2 x y^2 - 2 b^10 c^2 x y^2 +
3 a^8 c^4 x y^2 + 46 a^4 b^4 c^4 x y^2 - 17 b^8 c^4 x y^2 -
8 a^6 c^6 x y^2 + 4 a^4 b^2 c^6 x y^2 - 24 a^2 b^4 c^6 x y^2 +
36 b^6 c^6 x y^2 + a^4 c^8 x y^2 + 4 a^2 b^2 c^8 x y^2 -
17 b^4 c^8 x y^2 - 2 b^2 c^10 x y^2 + c^12 x y^2 - a^12 y^3 +
4 a^10 b^2 y^3 + 7 a^8 b^4 y^3 - 40 a^6 b^6 y^3 + 53 a^4 b^8 y^3 -
28 a^2 b^10 y^3 + 5 b^12 y^3 - 2 a^10 c^2 y^3 + 4 a^8 b^2 c^2 y^3 +
36 a^6 b^4 c^2 y^3 - 120 a^4 b^6 c^2 y^3 + 110 a^2 b^8 c^2 y^3 -
28 b^10 c^2 y^3 + a^8 c^4 y^3 - 8 a^6 b^2 c^4 y^3 +
42 a^4 b^4 c^4 y^3 - 120 a^2 b^6 c^4 y^3 + 53 b^8 c^4 y^3 +
4 a^6 c^6 y^3 - 8 a^4 b^2 c^6 y^3 + 36 a^2 b^4 c^6 y^3 -
40 b^6 c^6 y^3 + a^4 c^8 y^3 + 4 a^2 b^2 c^8 y^3 + 7 b^4 c^8 y^3 -
2 a^2 c^10 y^3 + 4 b^2 c^10 y^3 - c^12 y^3 + a^12 x^2 z -
2 a^10 b^2 x^2 z - 17 a^8 b^4 x^2 z + 36 a^6 b^6 x^2 z -
17 a^4 b^8 x^2 z - 2 a^2 b^10 x^2 z + b^12 x^2 z -
4 a^10 c^2 x^2 z + 24 a^8 b^2 c^2 x^2 z - 24 a^4 b^6 c^2 x^2 z +
4 a^2 b^8 c^2 x^2 z - 7 a^8 c^4 x^2 z - 44 a^6 b^2 c^4 x^2 z +
46 a^4 b^4 c^4 x^2 z + 4 a^2 b^6 c^4 x^2 z + b^8 c^4 x^2 z +
40 a^6 c^6 x^2 z + 48 a^4 b^2 c^6 x^2 z - 8 b^6 c^6 x^2 z -
53 a^4 c^8 x^2 z - 34 a^2 b^2 c^8 x^2 z + 3 b^4 c^8 x^2 z +
28 a^2 c^10 x^2 z + 8 b^2 c^10 x^2 z - 5 c^12 x^2 z +
10 a^12 x y z - 36 a^10 b^2 x y z + 22 a^8 b^4 x y z +
8 a^6 b^6 x y z + 22 a^4 b^8 x y z - 36 a^2 b^10 x y z +
10 b^12 x y z - 36 a^10 c^2 x y z + 124 a^8 b^2 c^2 x y z -
88 a^6 b^4 c^2 x y z - 88 a^4 b^6 c^2 x y z +
124 a^2 b^8 c^2 x y z - 36 b^10 c^2 x y z + 22 a^8 c^4 x y z -
88 a^6 b^2 c^4 x y z + 132 a^4 b^4 c^4 x y z -
88 a^2 b^6 c^4 x y z + 22 b^8 c^4 x y z + 8 a^6 c^6 x y z -
88 a^4 b^2 c^6 x y z - 88 a^2 b^4 c^6 x y z + 8 b^6 c^6 x y z +
22 a^4 c^8 x y z + 124 a^2 b^2 c^8 x y z + 22 b^4 c^8 x y z -
36 a^2 c^10 x y z - 36 b^2 c^10 x y z + 10 c^12 x y z + a^12 y^2 z -
2 a^10 b^2 y^2 z - 17 a^8 b^4 y^2 z + 36 a^6 b^6 y^2 z -
17 a^4 b^8 y^2 z - 2 a^2 b^10 y^2 z + b^12 y^2 z +
4 a^8 b^2 c^2 y^2 z - 24 a^6 b^4 c^2 y^2 z + 24 a^2 b^8 c^2 y^2 z -
4 b^10 c^2 y^2 z + a^8 c^4 y^2 z + 4 a^6 b^2 c^4 y^2 z +
46 a^4 b^4 c^4 y^2 z - 44 a^2 b^6 c^4 y^2 z - 7 b^8 c^4 y^2 z -
8 a^6 c^6 y^2 z + 48 a^2 b^4 c^6 y^2 z + 40 b^6 c^6 y^2 z +
3 a^4 c^8 y^2 z - 34 a^2 b^2 c^8 y^2 z - 53 b^4 c^8 y^2 z +
8 a^2 c^10 y^2 z + 28 b^2 c^10 y^2 z - 5 c^12 y^2 z - 5 a^12 x z^2 +
8 a^10 b^2 x z^2 + 3 a^8 b^4 x z^2 - 8 a^6 b^6 x z^2 +
a^4 b^8 x z^2 + b^12 x z^2 + 28 a^10 c^2 x z^2 -
34 a^8 b^2 c^2 x z^2 + 4 a^4 b^6 c^2 x z^2 + 4 a^2 b^8 c^2 x z^2 -
2 b^10 c^2 x z^2 - 53 a^8 c^4 x z^2 + 48 a^6 b^2 c^4 x z^2 +
46 a^4 b^4 c^4 x z^2 - 24 a^2 b^6 c^4 x z^2 - 17 b^8 c^4 x z^2 +
40 a^6 c^6 x z^2 - 44 a^4 b^2 c^6 x z^2 + 36 b^6 c^6 x z^2 -
7 a^4 c^8 x z^2 + 24 a^2 b^2 c^8 x z^2 - 17 b^4 c^8 x z^2 -
4 a^2 c^10 x z^2 - 2 b^2 c^10 x z^2 + c^12 x z^2 + a^12 y z^2 +
a^8 b^4 y z^2 - 8 a^6 b^6 y z^2 + 3 a^4 b^8 y z^2 +
8 a^2 b^10 y z^2 - 5 b^12 y z^2 - 2 a^10 c^2 y z^2 +
4 a^8 b^2 c^2 y z^2 + 4 a^6 b^4 c^2 y z^2 - 34 a^2 b^8 c^2 y z^2 +
28 b^10 c^2 y z^2 - 17 a^8 c^4 y z^2 - 24 a^6 b^2 c^4 y z^2 +
46 a^4 b^4 c^4 y z^2 + 48 a^2 b^6 c^4 y z^2 - 53 b^8 c^4 y z^2 +
36 a^6 c^6 y z^2 - 44 a^2 b^4 c^6 y z^2 + 40 b^6 c^6 y z^2 -
17 a^4 c^8 y z^2 + 24 a^2 b^2 c^8 y z^2 - 7 b^4 c^8 y z^2 -
2 a^2 c^10 y z^2 - 4 b^2 c^10 y z^2 + c^12 y z^2 - a^12 z^3 -
2 a^10 b^2 z^3 + a^8 b^4 z^3 + 4 a^6 b^6 z^3 + a^4 b^8 z^3 -
2 a^2 b^10 z^3 - b^12 z^3 + 4 a^10 c^2 z^3 + 4 a^8 b^2 c^2 z^3 -
8 a^6 b^4 c^2 z^3 - 8 a^4 b^6 c^2 z^3 + 4 a^2 b^8 c^2 z^3 +
4 b^10 c^2 z^3 + 7 a^8 c^4 z^3 + 36 a^6 b^2 c^4 z^3 +
42 a^4 b^4 c^4 z^3 + 36 a^2 b^6 c^4 z^3 + 7 b^8 c^4 z^3 -
40 a^6 c^6 z^3 - 120 a^4 b^2 c^6 z^3 - 120 a^2 b^4 c^6 z^3 -
40 b^6 c^6 z^3 + 53 a^4 c^8 z^3 + 110 a^2 b^2 c^8 z^3 +
53 b^4 c^8 z^3 - 28 a^2 c^10 z^3 - 28 b^2 c^10 z^3 + 5 c^12 z^3

Francisco Javier García Capitán
17 March 2012