## 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

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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