Reflections of a line through H in parallels to altitudes through A,B,C

Let ABC be a triangle, HaHbHc the orthic triangle, L a line through H and Lab,Lac the parallels through A to altitudes BHb, CHc, resp. The reflections of L in Lab, Lac intersect at A'. Similarly B',C'.

The triangles ABC, A'B'C' are perspective. The perspector lies on the circumcircle of ABC, where concur the reflections of L in the sidelines of ABC.

Note: The triangle A'B'C' is also perspective with the antimedial and medial triangles of ABC.

APH, 25 November 2011

Reflections in cevians : LOCUS

We can wonder if taking the isogonal conjugate of P here is natural, or, given P, which Q satisfy the same property that the isogonal conjugate P* in your statement.

[i.e. Let ABC be a triangle, P = (u:v:w) a fixed point, Q = (x:y:z) a variable point, A'B'C' the pedal triangle of P, La,Lb,Lc the parallels to PQ through A',B',C', resp. and L1,L2,L3 the reflections of La,Lb,Lc in the cevians AQ, BQ, CQ, resp. Which is the locus of Q such that L1,L2,L3 are concurrent?]

Will get a locus containing the isogonal conjugate of P:

The locus is a sextic throught P, P*, H, and the orthic triangle's vertices Ha,Hb,Hc.

For P = (u:v:w) the equation of the sextic is:

-a^4 b^2 c^4 u v x^4 y^2 + 2 a^2 b^4 c^4 u v x^4 y^2 -
b^6 c^4 u v x^4 y^2 + b^2 c^8 u v x^4 y^2 - a^6 c^4 v^2 x^4 y^2 +
2 a^4 b^2 c^4 v^2 x^4 y^2 - a^2 b^4 c^4 v^2 x^4 y^2 +
2 a^4 c^6 v^2 x^4 y^2 - 2 a^2 b^2 c^6 v^2 x^4 y^2 -
a^2 c^8 v^2 x^4 y^2 + a^4 b^2 c^4 u w x^4 y^2 -
b^6 c^4 u w x^4 y^2 - 2 a^2 b^2 c^6 u w x^4 y^2 +
b^2 c^8 u w x^4 y^2 + 2 a^4 b^2 c^4 v w x^4 y^2 -
2 a^2 b^4 c^4 v w x^4 y^2 - 2 a^2 b^2 c^6 v w x^4 y^2 +
a^4 b^2 c^4 u^2 x^3 y^3 - 2 a^2 b^4 c^4 u^2 x^3 y^3 +
b^6 c^4 u^2 x^3 y^3 - b^2 c^8 u^2 x^3 y^3 + a^6 c^4 u v x^3 y^3 -
3 a^4 b^2 c^4 u v x^3 y^3 + 3 a^2 b^4 c^4 u v x^3 y^3 -
b^6 c^4 u v x^3 y^3 - 2 a^4 c^6 u v x^3 y^3 + 2 b^4 c^6 u v x^3 y^3 +
a^2 c^8 u v x^3 y^3 - b^2 c^8 u v x^3 y^3 - a^6 c^4 v^2 x^3 y^3 +
2 a^4 b^2 c^4 v^2 x^3 y^3 - a^2 b^4 c^4 v^2 x^3 y^3 +
a^2 c^8 v^2 x^3 y^3 + 3 a^4 b^2 c^4 u w x^3 y^3 -
2 a^2 b^4 c^4 u w x^3 y^3 - b^6 c^4 u w x^3 y^3 -
2 a^2 b^2 c^6 u w x^3 y^3 + 2 b^4 c^6 u w x^3 y^3 -
b^2 c^8 u w x^3 y^3 + a^6 c^4 v w x^3 y^3 +
2 a^4 b^2 c^4 v w x^3 y^3 - 3 a^2 b^4 c^4 v w x^3 y^3 -
2 a^4 c^6 v w x^3 y^3 + 2 a^2 b^2 c^6 v w x^3 y^3 +
a^2 c^8 v w x^3 y^3 + a^4 b^2 c^4 u^2 x^2 y^4 -
2 a^2 b^4 c^4 u^2 x^2 y^4 + b^6 c^4 u^2 x^2 y^4 +
2 a^2 b^2 c^6 u^2 x^2 y^4 - 2 b^4 c^6 u^2 x^2 y^4 +
b^2 c^8 u^2 x^2 y^4 + a^6 c^4 u v x^2 y^4 -
2 a^4 b^2 c^4 u v x^2 y^4 + a^2 b^4 c^4 u v x^2 y^4 -
a^2 c^8 u v x^2 y^4 + 2 a^4 b^2 c^4 u w x^2 y^4 -
2 a^2 b^4 c^4 u w x^2 y^4 + 2 a^2 b^2 c^6 u w x^2 y^4 +
a^6 c^4 v w x^2 y^4 - a^2 b^4 c^4 v w x^2 y^4 +
2 a^2 b^2 c^6 v w x^2 y^4 - a^2 c^8 v w x^2 y^4 +
a^6 b^2 c^2 u v x^4 y z - 3 a^4 b^4 c^2 u v x^4 y z +
3 a^2 b^6 c^2 u v x^4 y z - b^8 c^2 u v x^4 y z -
a^4 b^2 c^4 u v x^4 y z + 2 a^2 b^4 c^4 u v x^4 y z -
b^6 c^4 u v x^4 y z - a^2 b^2 c^6 u v x^4 y z + b^4 c^6 u v x^4 y z +
b^2 c^8 u v x^4 y z + 2 a^4 b^2 c^4 v^2 x^4 y z -
2 a^2 b^4 c^4 v^2 x^4 y z - 2 a^2 b^2 c^6 v^2 x^4 y z -
a^6 b^2 c^2 u w x^4 y z + a^4 b^4 c^2 u w x^4 y z +
a^2 b^6 c^2 u w x^4 y z - b^8 c^2 u w x^4 y z +
3 a^4 b^2 c^4 u w x^4 y z - 2 a^2 b^4 c^4 u w x^4 y z -
b^6 c^4 u w x^4 y z - 3 a^2 b^2 c^6 u w x^4 y z +
b^4 c^6 u w x^4 y z + b^2 c^8 u w x^4 y z -
2 a^4 b^4 c^2 w^2 x^4 y z + 2 a^2 b^6 c^2 w^2 x^4 y z +
2 a^2 b^4 c^4 w^2 x^4 y z - a^6 b^2 c^2 u^2 x^3 y^2 z +
3 a^4 b^4 c^2 u^2 x^3 y^2 z - 3 a^2 b^6 c^2 u^2 x^3 y^2 z +
b^8 c^2 u^2 x^3 y^2 z - 2 a^2 b^4 c^4 u^2 x^3 y^2 z +
2 b^6 c^4 u^2 x^3 y^2 z + 3 a^2 b^2 c^6 u^2 x^3 y^2 z -
b^4 c^6 u^2 x^3 y^2 z - 2 b^2 c^8 u^2 x^3 y^2 z +
a^6 b^2 c^2 u v x^3 y^2 z - 3 a^4 b^4 c^2 u v x^3 y^2 z +
3 a^2 b^6 c^2 u v x^3 y^2 z - b^8 c^2 u v x^3 y^2 z -
6 a^4 b^2 c^4 u v x^3 y^2 z + 4 a^2 b^4 c^4 u v x^3 y^2 z +
2 b^6 c^4 u v x^3 y^2 z + 5 a^2 b^2 c^6 u v x^3 y^2 z -
b^4 c^6 u v x^3 y^2 z - a^8 c^2 v^2 x^3 y^2 z +
3 a^6 b^2 c^2 v^2 x^3 y^2 z - 3 a^4 b^4 c^2 v^2 x^3 y^2 z +
a^2 b^6 c^2 v^2 x^3 y^2 z + a^6 c^4 v^2 x^3 y^2 z -
2 a^4 b^2 c^4 v^2 x^3 y^2 z + a^2 b^4 c^4 v^2 x^3 y^2 z +
a^4 c^6 v^2 x^3 y^2 z - a^2 b^2 c^6 v^2 x^3 y^2 z -
a^2 c^8 v^2 x^3 y^2 z - 3 a^6 b^2 c^2 u w x^3 y^2 z +
5 a^4 b^4 c^2 u w x^3 y^2 z - a^2 b^6 c^2 u w x^3 y^2 z -
b^8 c^2 u w x^3 y^2 z + 6 a^4 b^2 c^4 u w x^3 y^2 z -
8 a^2 b^4 c^4 u w x^3 y^2 z + 2 b^6 c^4 u w x^3 y^2 z -
3 a^2 b^2 c^6 u w x^3 y^2 z - b^4 c^6 u w x^3 y^2 z +
2 a^6 b^2 c^2 v w x^3 y^2 z - 4 a^4 b^4 c^2 v w x^3 y^2 z +
2 a^2 b^6 c^2 v w x^3 y^2 z - 2 a^2 b^2 c^6 v w x^3 y^2 z -
2 a^6 b^2 c^2 w^2 x^3 y^2 z - 2 a^4 b^4 c^2 w^2 x^3 y^2 z +
4 a^2 b^6 c^2 w^2 x^3 y^2 z + 4 a^4 b^2 c^4 w^2 x^3 y^2 z -
2 a^2 b^4 c^4 w^2 x^3 y^2 z - 2 a^2 b^2 c^6 w^2 x^3 y^2 z -
a^6 b^2 c^2 u^2 x^2 y^3 z + 3 a^4 b^4 c^2 u^2 x^2 y^3 z -
3 a^2 b^6 c^2 u^2 x^2 y^3 z + b^8 c^2 u^2 x^2 y^3 z -
a^4 b^2 c^4 u^2 x^2 y^3 z + 2 a^2 b^4 c^4 u^2 x^2 y^3 z -
b^6 c^4 u^2 x^2 y^3 z + a^2 b^2 c^6 u^2 x^2 y^3 z -
b^4 c^6 u^2 x^2 y^3 z + b^2 c^8 u^2 x^2 y^3 z +
a^8 c^2 u v x^2 y^3 z - 3 a^6 b^2 c^2 u v x^2 y^3 z +
3 a^4 b^4 c^2 u v x^2 y^3 z - a^2 b^6 c^2 u v x^2 y^3 z -
2 a^6 c^4 u v x^2 y^3 z - 4 a^4 b^2 c^4 u v x^2 y^3 z +
6 a^2 b^4 c^4 u v x^2 y^3 z + a^4 c^6 u v x^2 y^3 z -
5 a^2 b^2 c^6 u v x^2 y^3 z - a^8 c^2 v^2 x^2 y^3 z +
3 a^6 b^2 c^2 v^2 x^2 y^3 z - 3 a^4 b^4 c^2 v^2 x^2 y^3 z +
a^2 b^6 c^2 v^2 x^2 y^3 z - 2 a^6 c^4 v^2 x^2 y^3 z +
2 a^4 b^2 c^4 v^2 x^2 y^3 z + a^4 c^6 v^2 x^2 y^3 z -
3 a^2 b^2 c^6 v^2 x^2 y^3 z + 2 a^2 c^8 v^2 x^2 y^3 z -
2 a^6 b^2 c^2 u w x^2 y^3 z + 4 a^4 b^4 c^2 u w x^2 y^3 z -
2 a^2 b^6 c^2 u w x^2 y^3 z + 2 a^2 b^2 c^6 u w x^2 y^3 z +
a^8 c^2 v w x^2 y^3 z + a^6 b^2 c^2 v w x^2 y^3 z -
5 a^4 b^4 c^2 v w x^2 y^3 z + 3 a^2 b^6 c^2 v w x^2 y^3 z -
2 a^6 c^4 v w x^2 y^3 z + 8 a^4 b^2 c^4 v w x^2 y^3 z -
6 a^2 b^4 c^4 v w x^2 y^3 z + a^4 c^6 v w x^2 y^3 z +
3 a^2 b^2 c^6 v w x^2 y^3 z - 4 a^6 b^2 c^2 w^2 x^2 y^3 z +
2 a^4 b^4 c^2 w^2 x^2 y^3 z + 2 a^2 b^6 c^2 w^2 x^2 y^3 z +
2 a^4 b^2 c^4 w^2 x^2 y^3 z - 4 a^2 b^4 c^4 w^2 x^2 y^3 z +
2 a^2 b^2 c^6 w^2 x^2 y^3 z + 2 a^4 b^2 c^4 u^2 x y^4 z -
2 a^2 b^4 c^4 u^2 x y^4 z + 2 a^2 b^2 c^6 u^2 x y^4 z +
a^8 c^2 u v x y^4 z - 3 a^6 b^2 c^2 u v x y^4 z +
3 a^4 b^4 c^2 u v x y^4 z - a^2 b^6 c^2 u v x y^4 z +
a^6 c^4 u v x y^4 z - 2 a^4 b^2 c^4 u v x y^4 z +
a^2 b^4 c^4 u v x y^4 z - a^4 c^6 u v x y^4 z +
a^2 b^2 c^6 u v x y^4 z - a^2 c^8 u v x y^4 z + a^8 c^2 v w x y^4 z -
a^6 b^2 c^2 v w x y^4 z - a^4 b^4 c^2 v w x y^4 z +
a^2 b^6 c^2 v w x y^4 z + a^6 c^4 v w x y^4 z +
2 a^4 b^2 c^4 v w x y^4 z - 3 a^2 b^4 c^4 v w x y^4 z -
a^4 c^6 v w x y^4 z + 3 a^2 b^2 c^6 v w x y^4 z -
a^2 c^8 v w x y^4 z - 2 a^6 b^2 c^2 w^2 x y^4 z +
2 a^4 b^4 c^2 w^2 x y^4 z - 2 a^4 b^2 c^4 w^2 x y^4 z -
a^4 b^4 c^2 u v x^4 z^2 + 2 a^2 b^6 c^2 u v x^4 z^2 -
b^8 c^2 u v x^4 z^2 + b^4 c^6 u v x^4 z^2 + a^4 b^4 c^2 u w x^4 z^2 -
b^8 c^2 u w x^4 z^2 - 2 a^2 b^4 c^4 u w x^4 z^2 +
b^4 c^6 u w x^4 z^2 - 2 a^4 b^4 c^2 v w x^4 z^2 +
2 a^2 b^6 c^2 v w x^4 z^2 + 2 a^2 b^4 c^4 v w x^4 z^2 +
a^6 b^4 w^2 x^4 z^2 - 2 a^4 b^6 w^2 x^4 z^2 + a^2 b^8 w^2 x^4 z^2 -
2 a^4 b^4 c^2 w^2 x^4 z^2 + 2 a^2 b^6 c^2 w^2 x^4 z^2 +
a^2 b^4 c^4 w^2 x^4 z^2 + a^6 b^2 c^2 u^2 x^3 y z^2 -
3 a^2 b^6 c^2 u^2 x^3 y z^2 + 2 b^8 c^2 u^2 x^3 y z^2 -
3 a^4 b^2 c^4 u^2 x^3 y z^2 + 2 a^2 b^4 c^4 u^2 x^3 y z^2 +
b^6 c^4 u^2 x^3 y z^2 + 3 a^2 b^2 c^6 u^2 x^3 y z^2 -
2 b^4 c^6 u^2 x^3 y z^2 - b^2 c^8 u^2 x^3 y z^2 +
3 a^6 b^2 c^2 u v x^3 y z^2 - 6 a^4 b^4 c^2 u v x^3 y z^2 +
3 a^2 b^6 c^2 u v x^3 y z^2 - 5 a^4 b^2 c^4 u v x^3 y z^2 +
8 a^2 b^4 c^4 u v x^3 y z^2 + b^6 c^4 u v x^3 y z^2 +
a^2 b^2 c^6 u v x^3 y z^2 - 2 b^4 c^6 u v x^3 y z^2 +
b^2 c^8 u v x^3 y z^2 + 2 a^6 b^2 c^2 v^2 x^3 y z^2 -
4 a^4 b^4 c^2 v^2 x^3 y z^2 + 2 a^2 b^6 c^2 v^2 x^3 y z^2 +
2 a^4 b^2 c^4 v^2 x^3 y z^2 + 2 a^2 b^4 c^4 v^2 x^3 y z^2 -
4 a^2 b^2 c^6 v^2 x^3 y z^2 - a^6 b^2 c^2 u w x^3 y z^2 +
6 a^4 b^4 c^2 u w x^3 y z^2 - 5 a^2 b^6 c^2 u w x^3 y z^2 +
3 a^4 b^2 c^4 u w x^3 y z^2 - 4 a^2 b^4 c^4 u w x^3 y z^2 +
b^6 c^4 u w x^3 y z^2 - 3 a^2 b^2 c^6 u w x^3 y z^2 -
2 b^4 c^6 u w x^3 y z^2 + b^2 c^8 u w x^3 y z^2 -
2 a^6 b^2 c^2 v w x^3 y z^2 + 2 a^2 b^6 c^2 v w x^3 y z^2 +
4 a^4 b^2 c^4 v w x^3 y z^2 - 2 a^2 b^2 c^6 v w x^3 y z^2 +
a^8 b^2 w^2 x^3 y z^2 - a^6 b^4 w^2 x^3 y z^2 -
a^4 b^6 w^2 x^3 y z^2 + a^2 b^8 w^2 x^3 y z^2 -
3 a^6 b^2 c^2 w^2 x^3 y z^2 + 2 a^4 b^4 c^2 w^2 x^3 y z^2 +
a^2 b^6 c^2 w^2 x^3 y z^2 + 3 a^4 b^2 c^4 w^2 x^3 y z^2 -
a^2 b^4 c^4 w^2 x^3 y z^2 - a^2 b^2 c^6 w^2 x^3 y z^2 +
a^4 b^4 c^2 u^2 x^2 y^2 z^2 - 2 a^2 b^6 c^2 u^2 x^2 y^2 z^2 +
b^8 c^2 u^2 x^2 y^2 z^2 - a^4 b^2 c^4 u^2 x^2 y^2 z^2 -
3 b^6 c^4 u^2 x^2 y^2 z^2 + 2 a^2 b^2 c^6 u^2 x^2 y^2 z^2 +
3 b^4 c^6 u^2 x^2 y^2 z^2 - b^2 c^8 u^2 x^2 y^2 z^2 -
a^8 c^2 v^2 x^2 y^2 z^2 + 2 a^6 b^2 c^2 v^2 x^2 y^2 z^2 -
a^4 b^4 c^2 v^2 x^2 y^2 z^2 + 3 a^6 c^4 v^2 x^2 y^2 z^2 +
a^2 b^4 c^4 v^2 x^2 y^2 z^2 - 3 a^4 c^6 v^2 x^2 y^2 z^2 -
2 a^2 b^2 c^6 v^2 x^2 y^2 z^2 + a^2 c^8 v^2 x^2 y^2 z^2 +
a^8 b^2 w^2 x^2 y^2 z^2 - 3 a^6 b^4 w^2 x^2 y^2 z^2 +
3 a^4 b^6 w^2 x^2 y^2 z^2 - a^2 b^8 w^2 x^2 y^2 z^2 -
2 a^6 b^2 c^2 w^2 x^2 y^2 z^2 + 2 a^2 b^6 c^2 w^2 x^2 y^2 z^2 +
a^4 b^2 c^4 w^2 x^2 y^2 z^2 - a^2 b^4 c^4 w^2 x^2 y^2 z^2 -
2 a^6 b^2 c^2 u^2 x y^3 z^2 + 4 a^4 b^4 c^2 u^2 x y^3 z^2 -
2 a^2 b^6 c^2 u^2 x y^3 z^2 - 2 a^4 b^2 c^4 u^2 x y^3 z^2 -
2 a^2 b^4 c^4 u^2 x y^3 z^2 + 4 a^2 b^2 c^6 u^2 x y^3 z^2 -
3 a^6 b^2 c^2 u v x y^3 z^2 + 6 a^4 b^4 c^2 u v x y^3 z^2 -
3 a^2 b^6 c^2 u v x y^3 z^2 - a^6 c^4 u v x y^3 z^2 -
8 a^4 b^2 c^4 u v x y^3 z^2 + 5 a^2 b^4 c^4 u v x y^3 z^2 +
2 a^4 c^6 u v x y^3 z^2 - a^2 b^2 c^6 u v x y^3 z^2 -
a^2 c^8 u v x y^3 z^2 - 2 a^8 c^2 v^2 x y^3 z^2 +
3 a^6 b^2 c^2 v^2 x y^3 z^2 - a^2 b^6 c^2 v^2 x y^3 z^2 -
a^6 c^4 v^2 x y^3 z^2 - 2 a^4 b^2 c^4 v^2 x y^3 z^2 +
3 a^2 b^4 c^4 v^2 x y^3 z^2 + 2 a^4 c^6 v^2 x y^3 z^2 -
3 a^2 b^2 c^6 v^2 x y^3 z^2 + a^2 c^8 v^2 x y^3 z^2 -
2 a^6 b^2 c^2 u w x y^3 z^2 + 2 a^2 b^6 c^2 u w x y^3 z^2 -
4 a^2 b^4 c^4 u w x y^3 z^2 + 2 a^2 b^2 c^6 u w x y^3 z^2 +
5 a^6 b^2 c^2 v w x y^3 z^2 - 6 a^4 b^4 c^2 v w x y^3 z^2 +
a^2 b^6 c^2 v w x y^3 z^2 - a^6 c^4 v w x y^3 z^2 +
4 a^4 b^2 c^4 v w x y^3 z^2 - 3 a^2 b^4 c^4 v w x y^3 z^2 +
2 a^4 c^6 v w x y^3 z^2 + 3 a^2 b^2 c^6 v w x y^3 z^2 -
a^2 c^8 v w x y^3 z^2 - a^8 b^2 w^2 x y^3 z^2 +
a^6 b^4 w^2 x y^3 z^2 + a^4 b^6 w^2 x y^3 z^2 -
a^2 b^8 w^2 x y^3 z^2 - a^6 b^2 c^2 w^2 x y^3 z^2 -
2 a^4 b^4 c^2 w^2 x y^3 z^2 + 3 a^2 b^6 c^2 w^2 x y^3 z^2 +
a^4 b^2 c^4 w^2 x y^3 z^2 - 3 a^2 b^4 c^4 w^2 x y^3 z^2 +
a^2 b^2 c^6 w^2 x y^3 z^2 + a^8 c^2 u v y^4 z^2 -
2 a^6 b^2 c^2 u v y^4 z^2 + a^4 b^4 c^2 u v y^4 z^2 -
a^4 c^6 u v y^4 z^2 - 2 a^6 b^2 c^2 u w y^4 z^2 +
2 a^4 b^4 c^2 u w y^4 z^2 - 2 a^4 b^2 c^4 u w y^4 z^2 +
a^8 c^2 v w y^4 z^2 - a^4 b^4 c^2 v w y^4 z^2 +
2 a^4 b^2 c^4 v w y^4 z^2 - a^4 c^6 v w y^4 z^2 -
a^8 b^2 w^2 y^4 z^2 + 2 a^6 b^4 w^2 y^4 z^2 - a^4 b^6 w^2 y^4 z^2 -
2 a^6 b^2 c^2 w^2 y^4 z^2 + 2 a^4 b^4 c^2 w^2 y^4 z^2 -
a^4 b^2 c^4 w^2 y^4 z^2 - a^4 b^4 c^2 u^2 x^3 z^3 +
b^8 c^2 u^2 x^3 z^3 + 2 a^2 b^4 c^4 u^2 x^3 z^3 -
b^4 c^6 u^2 x^3 z^3 - 3 a^4 b^4 c^2 u v x^3 z^3 +
2 a^2 b^6 c^2 u v x^3 z^3 + b^8 c^2 u v x^3 z^3 +
2 a^2 b^4 c^4 u v x^3 z^3 - 2 b^6 c^4 u v x^3 z^3 +
b^4 c^6 u v x^3 z^3 - a^6 b^4 u w x^3 z^3 + 2 a^4 b^6 u w x^3 z^3 -
a^2 b^8 u w x^3 z^3 + 3 a^4 b^4 c^2 u w x^3 z^3 +
b^8 c^2 u w x^3 z^3 - 3 a^2 b^4 c^4 u w x^3 z^3 -
2 b^6 c^4 u w x^3 z^3 + b^4 c^6 u w x^3 z^3 - a^6 b^4 v w x^3 z^3 +
2 a^4 b^6 v w x^3 z^3 - a^2 b^8 v w x^3 z^3 -
2 a^4 b^4 c^2 v w x^3 z^3 - 2 a^2 b^6 c^2 v w x^3 z^3 +
3 a^2 b^4 c^4 v w x^3 z^3 + a^6 b^4 w^2 x^3 z^3 -
a^2 b^8 w^2 x^3 z^3 - 2 a^4 b^4 c^2 w^2 x^3 z^3 +
a^2 b^4 c^4 w^2 x^3 z^3 + a^6 b^2 c^2 u^2 x^2 y z^3 +
a^4 b^4 c^2 u^2 x^2 y z^3 - a^2 b^6 c^2 u^2 x^2 y z^3 -
b^8 c^2 u^2 x^2 y z^3 - 3 a^4 b^2 c^4 u^2 x^2 y z^3 -
2 a^2 b^4 c^4 u^2 x^2 y z^3 + b^6 c^4 u^2 x^2 y z^3 +
3 a^2 b^2 c^6 u^2 x^2 y z^3 + b^4 c^6 u^2 x^2 y z^3 -
b^2 c^8 u^2 x^2 y z^3 + 2 a^6 b^2 c^2 u v x^2 y z^3 -
2 a^2 b^6 c^2 u v x^2 y z^3 - 4 a^4 b^2 c^4 u v x^2 y z^3 +
2 a^2 b^2 c^6 u v x^2 y z^3 + 4 a^6 b^2 c^2 v^2 x^2 y z^3 -
2 a^4 b^4 c^2 v^2 x^2 y z^3 - 2 a^2 b^6 c^2 v^2 x^2 y z^3 -
2 a^4 b^2 c^4 v^2 x^2 y z^3 + 4 a^2 b^4 c^4 v^2 x^2 y z^3 -
2 a^2 b^2 c^6 v^2 x^2 y z^3 - a^8 b^2 u w x^2 y z^3 +
2 a^6 b^4 u w x^2 y z^3 - a^4 b^6 u w x^2 y z^3 +
3 a^6 b^2 c^2 u w x^2 y z^3 + 4 a^4 b^4 c^2 u w x^2 y z^3 +
5 a^2 b^6 c^2 u w x^2 y z^3 - 3 a^4 b^2 c^4 u w x^2 y z^3 -
6 a^2 b^4 c^4 u w x^2 y z^3 + a^2 b^2 c^6 u w x^2 y z^3 -
a^8 b^2 v w x^2 y z^3 + 2 a^6 b^4 v w x^2 y z^3 -
a^4 b^6 v w x^2 y z^3 - a^6 b^2 c^2 v w x^2 y z^3 -
8 a^4 b^4 c^2 v w x^2 y z^3 - 3 a^2 b^6 c^2 v w x^2 y z^3 +
5 a^4 b^2 c^4 v w x^2 y z^3 + 6 a^2 b^4 c^4 v w x^2 y z^3 -
3 a^2 b^2 c^6 v w x^2 y z^3 + a^8 b^2 w^2 x^2 y z^3 +
2 a^6 b^4 w^2 x^2 y z^3 - a^4 b^6 w^2 x^2 y z^3 -
2 a^2 b^8 w^2 x^2 y z^3 - 3 a^6 b^2 c^2 w^2 x^2 y z^3 -
2 a^4 b^4 c^2 w^2 x^2 y z^3 + 3 a^2 b^6 c^2 w^2 x^2 y z^3 +
3 a^4 b^2 c^4 w^2 x^2 y z^3 - a^2 b^2 c^6 w^2 x^2 y z^3 +
2 a^6 b^2 c^2 u^2 x y^2 z^3 + 2 a^4 b^4 c^2 u^2 x y^2 z^3 -
4 a^2 b^6 c^2 u^2 x y^2 z^3 - 4 a^4 b^2 c^4 u^2 x y^2 z^3 +
2 a^2 b^4 c^4 u^2 x y^2 z^3 + 2 a^2 b^2 c^6 u^2 x y^2 z^3 +
2 a^6 b^2 c^2 u v x y^2 z^3 - 2 a^2 b^6 c^2 u v x y^2 z^3 +
4 a^2 b^4 c^4 u v x y^2 z^3 - 2 a^2 b^2 c^6 u v x y^2 z^3 +
a^8 c^2 v^2 x y^2 z^3 + a^6 b^2 c^2 v^2 x y^2 z^3 -
a^4 b^4 c^2 v^2 x y^2 z^3 - a^2 b^6 c^2 v^2 x y^2 z^3 -
a^6 c^4 v^2 x y^2 z^3 + 2 a^4 b^2 c^4 v^2 x y^2 z^3 +
3 a^2 b^4 c^4 v^2 x y^2 z^3 - a^4 c^6 v^2 x y^2 z^3 -
3 a^2 b^2 c^6 v^2 x y^2 z^3 + a^2 c^8 v^2 x y^2 z^3 +
a^6 b^4 u w x y^2 z^3 - 2 a^4 b^6 u w x y^2 z^3 +
a^2 b^8 u w x y^2 z^3 + 3 a^6 b^2 c^2 u w x y^2 z^3 +
8 a^4 b^4 c^2 u w x y^2 z^3 + a^2 b^6 c^2 u w x y^2 z^3 -
6 a^4 b^2 c^4 u w x y^2 z^3 - 5 a^2 b^4 c^4 u w x y^2 z^3 +
3 a^2 b^2 c^6 u w x y^2 z^3 + a^6 b^4 v w x y^2 z^3 -
2 a^4 b^6 v w x y^2 z^3 + a^2 b^8 v w x y^2 z^3 -
5 a^6 b^2 c^2 v w x y^2 z^3 - 4 a^4 b^4 c^2 v w x y^2 z^3 -
3 a^2 b^6 c^2 v w x y^2 z^3 + 6 a^4 b^2 c^4 v w x y^2 z^3 +
3 a^2 b^4 c^4 v w x y^2 z^3 - a^2 b^2 c^6 v w x y^2 z^3 +
2 a^8 b^2 w^2 x y^2 z^3 + a^6 b^4 w^2 x y^2 z^3 -
2 a^4 b^6 w^2 x y^2 z^3 - a^2 b^8 w^2 x y^2 z^3 -
3 a^6 b^2 c^2 w^2 x y^2 z^3 + 2 a^4 b^4 c^2 w^2 x y^2 z^3 +
3 a^2 b^6 c^2 w^2 x y^2 z^3 - 3 a^2 b^4 c^4 w^2 x y^2 z^3 +
a^2 b^2 c^6 w^2 x y^2 z^3 - a^8 c^2 u v y^3 z^3 -
2 a^6 b^2 c^2 u v y^3 z^3 + 3 a^4 b^4 c^2 u v y^3 z^3 +
2 a^6 c^4 u v y^3 z^3 - 2 a^4 b^2 c^4 u v y^3 z^3 -
a^4 c^6 u v y^3 z^3 - a^8 c^2 v^2 y^3 z^3 +
a^4 b^4 c^2 v^2 y^3 z^3 - 2 a^4 b^2 c^4 v^2 y^3 z^3 +
a^4 c^6 v^2 y^3 z^3 + a^8 b^2 u w y^3 z^3 - 2 a^6 b^4 u w y^3 z^3 +
a^4 b^6 u w y^3 z^3 + 2 a^6 b^2 c^2 u w y^3 z^3 +
2 a^4 b^4 c^2 u w y^3 z^3 - 3 a^4 b^2 c^4 u w y^3 z^3 +
a^8 b^2 v w y^3 z^3 - 2 a^6 b^4 v w y^3 z^3 + a^4 b^6 v w y^3 z^3 -
a^8 c^2 v w y^3 z^3 - 3 a^4 b^4 c^2 v w y^3 z^3 +
2 a^6 c^4 v w y^3 z^3 + 3 a^4 b^2 c^4 v w y^3 z^3 -
a^4 c^6 v w y^3 z^3 + a^8 b^2 w^2 y^3 z^3 - a^4 b^6 w^2 y^3 z^3 +
2 a^4 b^4 c^2 w^2 y^3 z^3 - a^4 b^2 c^4 w^2 y^3 z^3 -
a^4 b^4 c^2 u^2 x^2 z^4 - 2 a^2 b^6 c^2 u^2 x^2 z^4 -
b^8 c^2 u^2 x^2 z^4 + 2 a^2 b^4 c^4 u^2 x^2 z^4 +
2 b^6 c^4 u^2 x^2 z^4 - b^4 c^6 u^2 x^2 z^4 -
2 a^4 b^4 c^2 u v x^2 z^4 - 2 a^2 b^6 c^2 u v x^2 z^4 +
2 a^2 b^4 c^4 u v x^2 z^4 - a^6 b^4 u w x^2 z^4 +
a^2 b^8 u w x^2 z^4 + 2 a^4 b^4 c^2 u w x^2 z^4 -
a^2 b^4 c^4 u w x^2 z^4 - a^6 b^4 v w x^2 z^4 +
a^2 b^8 v w x^2 z^4 - 2 a^2 b^6 c^2 v w x^2 z^4 +
a^2 b^4 c^4 v w x^2 z^4 - 2 a^4 b^4 c^2 u^2 x y z^4 -
2 a^2 b^6 c^2 u^2 x y z^4 + 2 a^2 b^4 c^4 u^2 x y z^4 +
2 a^6 b^2 c^2 v^2 x y z^4 + 2 a^4 b^4 c^2 v^2 x y z^4 -
2 a^4 b^2 c^4 v^2 x y z^4 - a^8 b^2 u w x y z^4 -
a^6 b^4 u w x y z^4 + a^4 b^6 u w x y z^4 + a^2 b^8 u w x y z^4 +
3 a^6 b^2 c^2 u w x y z^4 + 2 a^4 b^4 c^2 u w x y z^4 -
a^2 b^6 c^2 u w x y z^4 - 3 a^4 b^2 c^4 u w x y z^4 -
a^2 b^4 c^4 u w x y z^4 + a^2 b^2 c^6 u w x y z^4 -
a^8 b^2 v w x y z^4 - a^6 b^4 v w x y z^4 + a^4 b^6 v w x y z^4 +
a^2 b^8 v w x y z^4 + a^6 b^2 c^2 v w x y z^4 -
2 a^4 b^4 c^2 v w x y z^4 - 3 a^2 b^6 c^2 v w x y z^4 +
a^4 b^2 c^4 v w x y z^4 + 3 a^2 b^4 c^4 v w x y z^4 -
a^2 b^2 c^6 v w x y z^4 + 2 a^6 b^2 c^2 u v y^2 z^4 +
2 a^4 b^4 c^2 u v y^2 z^4 - 2 a^4 b^2 c^4 u v y^2 z^4 +
a^8 c^2 v^2 y^2 z^4 + 2 a^6 b^2 c^2 v^2 y^2 z^4 +
a^4 b^4 c^2 v^2 y^2 z^4 - 2 a^6 c^4 v^2 y^2 z^4 -
2 a^4 b^2 c^4 v^2 y^2 z^4 + a^4 c^6 v^2 y^2 z^4 -
a^8 b^2 u w y^2 z^4 + a^4 b^6 u w y^2 z^4 +
2 a^6 b^2 c^2 u w y^2 z^4 - a^4 b^2 c^4 u w y^2 z^4 -
a^8 b^2 v w y^2 z^4 + a^4 b^6 v w y^2 z^4 -
2 a^4 b^4 c^2 v w y^2 z^4 + a^4 b^2 c^4 v w y^2 z^4

The figure is the case for P = G.

Francisco Javier García Capitán
21 November 2011

Reflections in cevians

Let ABC be a triangle, P,P* two isogonal conjugate points, A'B'C' the pedal triangle of P and La,Lb,Lc the parallels to PP* through A',B',C', resp.

The reflections L1,L2,L3 of La,Lb,Lc in the cevians AP*, BP*, CP*, resp. are concurrent.

Coordinates of the point of concurrence?
(for P = (x:y:z) in barycentrics)

APH, 19 November 2011

********

The point of concurrence R has long coordinates:

R ={a^2 (-b^4 c^8 x^6 y^3 + 3 a^2 b^2 c^8 x^4 y^5 + a^4 c^8 x^3 y^6 +
a^2 b^2 c^8 x^3 y^6 - a^2 c^10 x^3 y^6 + a^2 b^4 c^6 x^6 y^2 z -
2 b^6 c^6 x^6 y^2 z - b^4 c^8 x^6 y^2 z -
4 a^2 b^4 c^6 x^5 y^3 z - 4 a^4 b^2 c^6 x^4 y^4 z -
2 a^2 b^4 c^6 x^4 y^4 z + 4 a^2 b^2 c^8 x^4 y^4 z -
4 a^2 b^4 c^6 x^3 y^5 z + 4 a^2 b^2 c^8 x^3 y^5 z +
2 a^6 c^6 x^2 y^6 z - 2 a^4 b^2 c^6 x^2 y^6 z -
a^2 b^4 c^6 x^2 y^6 z - a^4 c^8 x^2 y^6 z +
2 a^2 b^2 c^8 x^2 y^6 z - a^2 c^10 x^2 y^6 z +
a^2 b^6 c^4 x^6 y z^2 - b^8 c^4 x^6 y z^2 - 2 b^6 c^6 x^6 y z^2 +
4 a^4 b^4 c^4 x^5 y^2 z^2 - 4 a^2 b^6 c^4 x^5 y^2 z^2 -
4 a^2 b^4 c^6 x^5 y^2 z^2 + 5 a^4 b^4 c^4 x^4 y^3 z^2 -
5 a^2 b^6 c^4 x^4 y^3 z^2 - 4 a^2 b^4 c^6 x^4 y^3 z^2 -
4 a^6 b^2 c^4 x^3 y^4 z^2 + 9 a^4 b^4 c^4 x^3 y^4 z^2 -
5 a^2 b^6 c^4 x^3 y^4 z^2 - 2 a^4 b^2 c^6 x^3 y^4 z^2 -
a^2 b^4 c^6 x^3 y^4 z^2 + 6 a^2 b^2 c^8 x^3 y^4 z^2 -
3 a^6 b^2 c^4 x^2 y^5 z^2 + 4 a^4 b^4 c^4 x^2 y^5 z^2 -
a^2 b^6 c^4 x^2 y^5 z^2 - 2 a^4 b^2 c^6 x^2 y^5 z^2 +
2 a^2 b^4 c^6 x^2 y^5 z^2 - a^2 b^2 c^8 x^2 y^5 z^2 +
a^8 c^4 x y^6 z^2 - 3 a^6 b^2 c^4 x y^6 z^2 +
2 a^4 b^4 c^4 x y^6 z^2 + a^6 c^6 x y^6 z^2 -
2 a^4 c^8 x y^6 z^2 - b^8 c^4 x^6 z^3 - 4 a^2 b^6 c^4 x^5 y z^3 +
5 a^4 b^4 c^4 x^4 y^2 z^3 - 4 a^2 b^6 c^4 x^4 y^2 z^3 -
5 a^2 b^4 c^6 x^4 y^2 z^3 + 16 a^4 b^4 c^4 x^3 y^3 z^3 -
4 a^6 b^2 c^4 x^2 y^4 z^3 + 19 a^4 b^4 c^4 x^2 y^4 z^3 +
2 a^2 b^6 c^4 x^2 y^4 z^3 + 2 a^4 b^2 c^6 x^2 y^4 z^3 -
4 a^2 b^4 c^6 x^2 y^4 z^3 + 2 a^2 b^2 c^8 x^2 y^4 z^3 -
4 a^6 b^2 c^4 x y^5 z^3 + 8 a^4 b^4 c^4 x y^5 z^3 -
8 a^4 b^2 c^6 x y^5 z^3 + a^8 c^4 y^6 z^3 -
2 a^6 b^2 c^4 y^6 z^3 - a^6 c^6 y^6 z^3 -
4 a^4 b^6 c^2 x^4 y z^4 + 4 a^2 b^8 c^2 x^4 y z^4 -
2 a^2 b^6 c^4 x^4 y z^4 - 4 a^6 b^4 c^2 x^3 y^2 z^4 -
2 a^4 b^6 c^2 x^3 y^2 z^4 + 6 a^2 b^8 c^2 x^3 y^2 z^4 +
9 a^4 b^4 c^4 x^3 y^2 z^4 - a^2 b^6 c^4 x^3 y^2 z^4 -
5 a^2 b^4 c^6 x^3 y^2 z^4 - 4 a^6 b^4 c^2 x^2 y^3 z^4 +
2 a^4 b^6 c^2 x^2 y^3 z^4 + 2 a^2 b^8 c^2 x^2 y^3 z^4 +
19 a^4 b^4 c^4 x^2 y^3 z^4 - 4 a^2 b^6 c^4 x^2 y^3 z^4 +
2 a^2 b^4 c^6 x^2 y^3 z^4 + 2 a^8 b^2 c^2 x y^4 z^4 -
2 a^6 b^4 c^2 x y^4 z^4 - 2 a^6 b^2 c^4 x y^4 z^4 +
2 a^8 b^2 c^2 y^5 z^4 - 2 a^6 b^4 c^2 y^5 z^4 -
7 a^6 b^2 c^4 y^5 z^4 + 3 a^2 b^8 c^2 x^4 z^5 +
4 a^2 b^8 c^2 x^3 y z^5 - 4 a^2 b^6 c^4 x^3 y z^5 -
3 a^6 b^4 c^2 x^2 y^2 z^5 - 2 a^4 b^6 c^2 x^2 y^2 z^5 -
a^2 b^8 c^2 x^2 y^2 z^5 + 4 a^4 b^4 c^4 x^2 y^2 z^5 +
2 a^2 b^6 c^4 x^2 y^2 z^5 - a^2 b^4 c^6 x^2 y^2 z^5 -
4 a^6 b^4 c^2 x y^3 z^5 - 8 a^4 b^6 c^2 x y^3 z^5 +
8 a^4 b^4 c^4 x y^3 z^5 + 2 a^8 b^2 c^2 y^4 z^5 -
7 a^6 b^4 c^2 y^4 z^5 - 2 a^6 b^2 c^4 y^4 z^5 + a^4 b^8 x^3 z^6 -
a^2 b^10 x^3 z^6 + a^2 b^8 c^2 x^3 z^6 + 2 a^6 b^6 x^2 y z^6 -
a^4 b^8 x^2 y z^6 - a^2 b^10 x^2 y z^6 -
2 a^4 b^6 c^2 x^2 y z^6 + 2 a^2 b^8 c^2 x^2 y z^6 -
a^2 b^6 c^4 x^2 y z^6 + a^8 b^4 x y^2 z^6 + a^6 b^6 x y^2 z^6 -
2 a^4 b^8 x y^2 z^6 - 3 a^6 b^4 c^2 x y^2 z^6 +
2 a^4 b^4 c^4 x y^2 z^6 + a^8 b^4 y^3 z^6 - a^6 b^6 y^3 z^6 -
2 a^6 b^4 c^2 y^3 z^6),
b^2 (a^2 b^2 c^8 x^6 y^3 + b^4 c^8 x^6 y^3 - b^2 c^10 x^6 y^3 +
3 a^2 b^2 c^8 x^5 y^4 - a^4 c^8 x^3 y^6 - a^4 b^2 c^6 x^6 y^2 z -
2 a^2 b^4 c^6 x^6 y^2 z + 2 b^6 c^6 x^6 y^2 z +
2 a^2 b^2 c^8 x^6 y^2 z - b^4 c^8 x^6 y^2 z -
b^2 c^10 x^6 y^2 z - 4 a^4 b^2 c^6 x^5 y^3 z +
4 a^2 b^2 c^8 x^5 y^3 z - 2 a^4 b^2 c^6 x^4 y^4 z -
4 a^2 b^4 c^6 x^4 y^4 z + 4 a^2 b^2 c^8 x^4 y^4 z -
4 a^4 b^2 c^6 x^3 y^5 z - 2 a^6 c^6 x^2 y^6 z +
a^4 b^2 c^6 x^2 y^6 z - a^4 c^8 x^2 y^6 z +
2 a^4 b^4 c^4 x^6 y z^2 - 3 a^2 b^6 c^4 x^6 y z^2 +
b^8 c^4 x^6 y z^2 + b^6 c^6 x^6 y z^2 - 2 b^4 c^8 x^6 y z^2 -
a^6 b^2 c^4 x^5 y^2 z^2 + 4 a^4 b^4 c^4 x^5 y^2 z^2 -
3 a^2 b^6 c^4 x^5 y^2 z^2 + 2 a^4 b^2 c^6 x^5 y^2 z^2 -
2 a^2 b^4 c^6 x^5 y^2 z^2 - a^2 b^2 c^8 x^5 y^2 z^2 -
5 a^6 b^2 c^4 x^4 y^3 z^2 + 9 a^4 b^4 c^4 x^4 y^3 z^2 -
4 a^2 b^6 c^4 x^4 y^3 z^2 - a^4 b^2 c^6 x^4 y^3 z^2 -
2 a^2 b^4 c^6 x^4 y^3 z^2 + 6 a^2 b^2 c^8 x^4 y^3 z^2 -
5 a^6 b^2 c^4 x^3 y^4 z^2 + 5 a^4 b^4 c^4 x^3 y^4 z^2 -
4 a^4 b^2 c^6 x^3 y^4 z^2 - 4 a^6 b^2 c^4 x^2 y^5 z^2 +
4 a^4 b^4 c^4 x^2 y^5 z^2 - 4 a^4 b^2 c^6 x^2 y^5 z^2 -
a^8 c^4 x y^6 z^2 + a^6 b^2 c^4 x y^6 z^2 - 2 a^6 c^6 x y^6 z^2 -
2 a^2 b^6 c^4 x^6 z^3 + b^8 c^4 x^6 z^3 - b^6 c^6 x^6 z^3 +
8 a^4 b^4 c^4 x^5 y z^3 - 4 a^2 b^6 c^4 x^5 y z^3 -
8 a^2 b^4 c^6 x^5 y z^3 + 2 a^6 b^2 c^4 x^4 y^2 z^3 +
19 a^4 b^4 c^4 x^4 y^2 z^3 - 4 a^2 b^6 c^4 x^4 y^2 z^3 -
4 a^4 b^2 c^6 x^4 y^2 z^3 + 2 a^2 b^4 c^6 x^4 y^2 z^3 +
2 a^2 b^2 c^8 x^4 y^2 z^3 + 16 a^4 b^4 c^4 x^3 y^3 z^3 -
4 a^6 b^2 c^4 x^2 y^4 z^3 + 5 a^4 b^4 c^4 x^2 y^4 z^3 -
5 a^4 b^2 c^6 x^2 y^4 z^3 - 4 a^6 b^2 c^4 x y^5 z^3 -
a^8 c^4 y^6 z^3 - 2 a^4 b^6 c^2 x^5 z^4 + 2 a^2 b^8 c^2 x^5 z^4 -
7 a^2 b^6 c^4 x^5 z^4 - 2 a^4 b^6 c^2 x^4 y z^4 +
2 a^2 b^8 c^2 x^4 y z^4 - 2 a^2 b^6 c^4 x^4 y z^4 +
2 a^8 b^2 c^2 x^3 y^2 z^4 + 2 a^6 b^4 c^2 x^3 y^2 z^4 -
4 a^4 b^6 c^2 x^3 y^2 z^4 - 4 a^6 b^2 c^4 x^3 y^2 z^4 +
19 a^4 b^4 c^4 x^3 y^2 z^4 + 2 a^4 b^2 c^6 x^3 y^2 z^4 +
6 a^8 b^2 c^2 x^2 y^3 z^4 - 2 a^6 b^4 c^2 x^2 y^3 z^4 -
4 a^4 b^6 c^2 x^2 y^3 z^4 - a^6 b^2 c^4 x^2 y^3 z^4 +
9 a^4 b^4 c^4 x^2 y^3 z^4 - 5 a^4 b^2 c^6 x^2 y^3 z^4 +
4 a^8 b^2 c^2 x y^4 z^4 - 4 a^6 b^4 c^2 x y^4 z^4 -
2 a^6 b^2 c^4 x y^4 z^4 - 7 a^4 b^6 c^2 x^4 z^5 +
2 a^2 b^8 c^2 x^4 z^5 - 2 a^2 b^6 c^4 x^4 z^5 -
8 a^6 b^4 c^2 x^3 y z^5 - 4 a^4 b^6 c^2 x^3 y z^5 +
8 a^4 b^4 c^4 x^3 y z^5 - a^8 b^2 c^2 x^2 y^2 z^5 -
2 a^6 b^4 c^2 x^2 y^2 z^5 - 3 a^4 b^6 c^2 x^2 y^2 z^5 +
2 a^6 b^2 c^4 x^2 y^2 z^5 + 4 a^4 b^4 c^4 x^2 y^2 z^5 -
a^4 b^2 c^6 x^2 y^2 z^5 + 4 a^8 b^2 c^2 x y^3 z^5 -
4 a^6 b^2 c^4 x y^3 z^5 + 3 a^8 b^2 c^2 y^4 z^5 -
a^6 b^6 x^3 z^6 + a^4 b^8 x^3 z^6 - 2 a^4 b^6 c^2 x^3 z^6 -
2 a^8 b^4 x^2 y z^6 + a^6 b^6 x^2 y z^6 + a^4 b^8 x^2 y z^6 -
3 a^4 b^6 c^2 x^2 y z^6 + 2 a^4 b^4 c^4 x^2 y z^6 -
a^10 b^2 x y^2 z^6 - a^8 b^4 x y^2 z^6 + 2 a^6 b^6 x y^2 z^6 +
2 a^8 b^2 c^2 x y^2 z^6 - 2 a^6 b^4 c^2 x y^2 z^6 -
a^6 b^2 c^4 x y^2 z^6 - a^10 b^2 y^3 z^6 + a^8 b^4 y^3 z^6 +
a^8 b^2 c^2 y^3 z^6),
c^2 (-2 a^2 b^4 c^6 x^6 y^3 - b^6 c^6 x^6 y^3 + b^4 c^8 x^6 y^3 -
2 a^4 b^2 c^6 x^5 y^4 - 7 a^2 b^4 c^6 x^5 y^4 +
2 a^2 b^2 c^8 x^5 y^4 - 7 a^4 b^2 c^6 x^4 y^5 -
2 a^2 b^4 c^6 x^4 y^5 + 2 a^2 b^2 c^8 x^4 y^5 - a^6 c^6 x^3 y^6 -
2 a^4 b^2 c^6 x^3 y^6 + a^4 c^8 x^3 y^6 +
2 a^4 b^4 c^4 x^6 y^2 z - 2 b^8 c^4 x^6 y^2 z -
3 a^2 b^4 c^6 x^6 y^2 z + b^6 c^6 x^6 y^2 z + b^4 c^8 x^6 y^2 z +
8 a^4 b^4 c^4 x^5 y^3 z - 8 a^2 b^6 c^4 x^5 y^3 z -
4 a^2 b^4 c^6 x^5 y^3 z - 2 a^4 b^2 c^6 x^4 y^4 z -
2 a^2 b^4 c^6 x^4 y^4 z + 2 a^2 b^2 c^8 x^4 y^4 z -
8 a^6 b^2 c^4 x^3 y^5 z + 8 a^4 b^4 c^4 x^3 y^5 z -
4 a^4 b^2 c^6 x^3 y^5 z - 2 a^8 c^4 x^2 y^6 z +
2 a^4 b^4 c^4 x^2 y^6 z + a^6 c^6 x^2 y^6 z -
3 a^4 b^2 c^6 x^2 y^6 z + a^4 c^8 x^2 y^6 z -
a^4 b^6 c^2 x^6 y z^2 + 2 a^2 b^8 c^2 x^6 y z^2 -
b^10 c^2 x^6 y z^2 - 2 a^2 b^6 c^4 x^6 y z^2 -
b^8 c^4 x^6 y z^2 + 2 b^6 c^6 x^6 y z^2 -
a^6 b^4 c^2 x^5 y^2 z^2 + 2 a^4 b^6 c^2 x^5 y^2 z^2 -
a^2 b^8 c^2 x^5 y^2 z^2 + 4 a^4 b^4 c^4 x^5 y^2 z^2 -
2 a^2 b^6 c^4 x^5 y^2 z^2 - 3 a^2 b^4 c^6 x^5 y^2 z^2 +
2 a^6 b^4 c^2 x^4 y^3 z^2 - 4 a^4 b^6 c^2 x^4 y^3 z^2 +
2 a^2 b^8 c^2 x^4 y^3 z^2 + 19 a^4 b^4 c^4 x^4 y^3 z^2 +
2 a^2 b^6 c^4 x^4 y^3 z^2 - 4 a^2 b^4 c^6 x^4 y^3 z^2 +
2 a^8 b^2 c^2 x^3 y^4 z^2 - 4 a^6 b^4 c^2 x^3 y^4 z^2 +
2 a^4 b^6 c^2 x^3 y^4 z^2 + 2 a^6 b^2 c^4 x^3 y^4 z^2 +
19 a^4 b^4 c^4 x^3 y^4 z^2 - 4 a^4 b^2 c^6 x^3 y^4 z^2 -
a^8 b^2 c^2 x^2 y^5 z^2 + 2 a^6 b^4 c^2 x^2 y^5 z^2 -
a^4 b^6 c^2 x^2 y^5 z^2 - 2 a^6 b^2 c^4 x^2 y^5 z^2 +
4 a^4 b^4 c^4 x^2 y^5 z^2 - 3 a^4 b^2 c^6 x^2 y^5 z^2 -
a^10 c^2 x y^6 z^2 + 2 a^8 b^2 c^2 x y^6 z^2 -
a^6 b^4 c^2 x y^6 z^2 - a^8 c^4 x y^6 z^2 -
2 a^6 b^2 c^4 x y^6 z^2 + 2 a^6 c^6 x y^6 z^2 +
a^2 b^8 c^2 x^6 z^3 - b^10 c^2 x^6 z^3 + b^8 c^4 x^6 z^3 -
4 a^4 b^6 c^2 x^5 y z^3 + 4 a^2 b^8 c^2 x^5 y z^3 -
5 a^6 b^4 c^2 x^4 y^2 z^3 - a^4 b^6 c^2 x^4 y^2 z^3 +
6 a^2 b^8 c^2 x^4 y^2 z^3 + 9 a^4 b^4 c^4 x^4 y^2 z^3 -
2 a^2 b^6 c^4 x^4 y^2 z^3 - 4 a^2 b^4 c^6 x^4 y^2 z^3 +
16 a^4 b^4 c^4 x^3 y^3 z^3 + 6 a^8 b^2 c^2 x^2 y^4 z^3 -
a^6 b^4 c^2 x^2 y^4 z^3 - 5 a^4 b^6 c^2 x^2 y^4 z^3 -
2 a^6 b^2 c^4 x^2 y^4 z^3 + 9 a^4 b^4 c^4 x^2 y^4 z^3 -
4 a^4 b^2 c^6 x^2 y^4 z^3 + 4 a^8 b^2 c^2 x y^5 z^3 -
4 a^6 b^4 c^2 x y^5 z^3 - a^10 c^2 y^6 z^3 + a^8 b^2 c^2 y^6 z^3 +
a^8 c^4 y^6 z^3 + 3 a^2 b^8 c^2 x^5 z^4 -
2 a^4 b^6 c^2 x^4 y z^4 + 4 a^2 b^8 c^2 x^4 y z^4 -
4 a^2 b^6 c^4 x^4 y z^4 - 5 a^6 b^4 c^2 x^3 y^2 z^4 -
4 a^4 b^6 c^2 x^3 y^2 z^4 + 5 a^4 b^4 c^4 x^3 y^2 z^4 -
4 a^6 b^4 c^2 x^2 y^3 z^4 - 5 a^4 b^6 c^2 x^2 y^3 z^4 +
5 a^4 b^4 c^4 x^2 y^3 z^4 + 4 a^8 b^2 c^2 x y^4 z^4 -
2 a^6 b^4 c^2 x y^4 z^4 - 4 a^6 b^2 c^4 x y^4 z^4 +
3 a^8 b^2 c^2 y^5 z^4 - 4 a^4 b^6 c^2 x^3 y z^5 -
4 a^6 b^4 c^2 x^2 y^2 z^5 - 4 a^4 b^6 c^2 x^2 y^2 z^5 +
4 a^4 b^4 c^4 x^2 y^2 z^5 - 4 a^6 b^4 c^2 x y^3 z^5 -
a^4 b^8 x^3 z^6 - 2 a^6 b^6 x^2 y z^6 - a^4 b^8 x^2 y z^6 +
a^4 b^6 c^2 x^2 y z^6 - a^8 b^4 x y^2 z^6 - 2 a^6 b^6 x y^2 z^6 +
a^6 b^4 c^2 x y^2 z^6 - a^8 b^4 y^3 z^6)}

The locus for this point being collinear with P and P*: when P is on circumcircle together with the 10th degree curve:

-a^2 b^4 c^8 x^7 y^3 - b^6 c^8 x^7 y^3 + b^4 c^10 x^7 y^3 -
4 a^2 b^4 c^8 x^6 y^4 + 4 a^4 b^2 c^8 x^4 y^6 + a^6 c^8 x^3 y^7 +
a^4 b^2 c^8 x^3 y^7 - a^4 c^10 x^3 y^7 + a^4 b^4 c^6 x^7 y^2 z +
a^2 b^6 c^6 x^7 y^2 z - 2 b^8 c^6 x^7 y^2 z -
2 a^2 b^4 c^8 x^7 y^2 z + b^6 c^8 x^7 y^2 z + b^4 c^10 x^7 y^2 z +
4 a^4 b^4 c^6 x^6 y^3 z - 4 a^2 b^6 c^6 x^6 y^3 z -
4 a^2 b^4 c^8 x^6 y^3 z - 3 a^4 b^4 c^6 x^5 y^4 z +
3 a^2 b^6 c^6 x^5 y^4 z - 3 a^2 b^4 c^8 x^5 y^4 z -
3 a^6 b^2 c^6 x^4 y^5 z + 3 a^4 b^4 c^6 x^4 y^5 z +
3 a^4 b^2 c^8 x^4 y^5 z + 4 a^6 b^2 c^6 x^3 y^6 z -
4 a^4 b^4 c^6 x^3 y^6 z + 4 a^4 b^2 c^8 x^3 y^6 z +
2 a^8 c^6 x^2 y^7 z - a^6 b^2 c^6 x^2 y^7 z - a^4 b^4 c^6 x^2 y^7 z -
a^6 c^8 x^2 y^7 z + 2 a^4 b^2 c^8 x^2 y^7 z - a^4 c^10 x^2 y^7 z -
a^4 b^6 c^4 x^7 y z^2 + 2 a^2 b^8 c^4 x^7 y z^2 -
b^10 c^4 x^7 y z^2 - a^2 b^6 c^6 x^7 y z^2 - b^8 c^6 x^7 y z^2 +
2 b^6 c^8 x^7 y z^2 + 3 a^6 b^4 c^4 x^5 y^3 z^2 -
6 a^4 b^6 c^4 x^5 y^3 z^2 + 3 a^2 b^8 c^4 x^5 y^3 z^2 +
3 a^4 b^4 c^6 x^5 y^3 z^2 + 3 a^2 b^6 c^6 x^5 y^3 z^2 -
6 a^2 b^4 c^8 x^5 y^3 z^2 - 3 a^8 b^2 c^4 x^3 y^5 z^2 +
6 a^6 b^4 c^4 x^3 y^5 z^2 - 3 a^4 b^6 c^4 x^3 y^5 z^2 -
3 a^6 b^2 c^6 x^3 y^5 z^2 - 3 a^4 b^4 c^6 x^3 y^5 z^2 +
6 a^4 b^2 c^8 x^3 y^5 z^2 + a^10 c^4 x y^7 z^2 -
2 a^8 b^2 c^4 x y^7 z^2 + a^6 b^4 c^4 x y^7 z^2 + a^8 c^6 x y^7 z^2 +
a^6 b^2 c^6 x y^7 z^2 - 2 a^6 c^8 x y^7 z^2 + a^2 b^8 c^4 x^7 z^3 -
b^10 c^4 x^7 z^3 + b^8 c^6 x^7 z^3 - 4 a^4 b^6 c^4 x^6 y z^3 +
4 a^2 b^8 c^4 x^6 y z^3 + 4 a^2 b^6 c^6 x^6 y z^3 -
3 a^6 b^4 c^4 x^5 y^2 z^3 - 3 a^4 b^6 c^4 x^5 y^2 z^3 +
6 a^2 b^8 c^4 x^5 y^2 z^3 + 6 a^4 b^4 c^6 x^5 y^2 z^3 -
3 a^2 b^6 c^6 x^5 y^2 z^3 - 3 a^2 b^4 c^8 x^5 y^2 z^3 -
6 a^8 b^2 c^4 x^2 y^5 z^3 + 3 a^6 b^4 c^4 x^2 y^5 z^3 +
3 a^4 b^6 c^4 x^2 y^5 z^3 + 3 a^6 b^2 c^6 x^2 y^5 z^3 -
6 a^4 b^4 c^6 x^2 y^5 z^3 + 3 a^4 b^2 c^8 x^2 y^5 z^3 -
4 a^8 b^2 c^4 x y^6 z^3 + 4 a^6 b^4 c^4 x y^6 z^3 -
4 a^6 b^2 c^6 x y^6 z^3 + a^10 c^4 y^7 z^3 - a^8 b^2 c^4 y^7 z^3 -
a^8 c^6 y^7 z^3 + 4 a^2 b^8 c^4 x^6 z^4 + 3 a^4 b^6 c^4 x^5 y z^4 +
3 a^2 b^8 c^4 x^5 y z^4 - 3 a^2 b^6 c^6 x^5 y z^4 -
3 a^8 b^2 c^4 x y^5 z^4 - 3 a^6 b^4 c^4 x y^5 z^4 +
3 a^6 b^2 c^6 x y^5 z^4 - 4 a^8 b^2 c^4 y^6 z^4 +
3 a^6 b^6 c^2 x^4 y z^5 - 3 a^4 b^8 c^2 x^4 y z^5 -
3 a^4 b^6 c^4 x^4 y z^5 + 3 a^8 b^4 c^2 x^3 y^2 z^5 +
3 a^6 b^6 c^2 x^3 y^2 z^5 - 6 a^4 b^8 c^2 x^3 y^2 z^5 -
6 a^6 b^4 c^4 x^3 y^2 z^5 + 3 a^4 b^6 c^4 x^3 y^2 z^5 +
3 a^4 b^4 c^6 x^3 y^2 z^5 + 6 a^8 b^4 c^2 x^2 y^3 z^5 -
3 a^6 b^6 c^2 x^2 y^3 z^5 - 3 a^4 b^8 c^2 x^2 y^3 z^5 -
3 a^6 b^4 c^4 x^2 y^3 z^5 + 6 a^4 b^6 c^4 x^2 y^3 z^5 -
3 a^4 b^4 c^6 x^2 y^3 z^5 + 3 a^8 b^4 c^2 x y^4 z^5 -
3 a^6 b^6 c^2 x y^4 z^5 + 3 a^6 b^4 c^4 x y^4 z^5 -
4 a^4 b^8 c^2 x^4 z^6 - 4 a^6 b^6 c^2 x^3 y z^6 -
4 a^4 b^8 c^2 x^3 y z^6 + 4 a^4 b^6 c^4 x^3 y z^6 +
4 a^8 b^4 c^2 x y^3 z^6 + 4 a^6 b^6 c^2 x y^3 z^6 -
4 a^6 b^4 c^4 x y^3 z^6 + 4 a^8 b^4 c^2 y^4 z^6 - a^6 b^8 x^3 z^7 +
a^4 b^10 x^3 z^7 - a^4 b^8 c^2 x^3 z^7 - 2 a^8 b^6 x^2 y z^7 +
a^6 b^8 x^2 y z^7 + a^4 b^10 x^2 y z^7 + a^6 b^6 c^2 x^2 y z^7 -
2 a^4 b^8 c^2 x^2 y z^7 + a^4 b^6 c^4 x^2 y z^7 -
a^10 b^4 x y^2 z^7 - a^8 b^6 x y^2 z^7 + 2 a^6 b^8 x y^2 z^7 +
2 a^8 b^4 c^2 x y^2 z^7 - a^6 b^6 c^2 x y^2 z^7 -
a^6 b^4 c^4 x y^2 z^7 - a^10 b^4 y^3 z^7 + a^8 b^6 y^3 z^7 +
a^8 b^4 c^2 y^3 z^7

The 10th degree curve contains the incenter and the excenters, but I didn't find other points.

Francisco Javier García Capitán
20 November 2011

LOCUS

ADDENDUM (9/9/19)
RELATED POINTS

X(34226) = MIDPOINT OF X(12149) AND X(15534)
X(34227) = X(111)X(15271)∩X(126)X(3258)
X(34228) = REFLECTION OF X(1367) IN X(1)

INRADIUS 5

Let ABC be a triangle and D a point on AB such that inradius of CDA = inradious of CDB := d.

Find the primitive heronian right triangles ABC, A = 90 d. (ie a^2 = b^2 + c^2, a,b,c integers, gcd(a,b,c) = 1) with d integer.

ABC primitive heronian right triangle with A = 90 d. ==>

a = x^2 + y^2

b or c = x^2 - y^2

c or b = 2xy

where x > y > 0 integers.

We have inradius r = y(x-y) and c = 2d^2 / (2d - r) [see HERE].

1. c = 2xy, b = x^2 - y^2

==> xy = d^2 /(2d - y(x-y)) ==> d = xy +-ysqrt(xy) and since x>y,

d = xy + ysqrt(xy).

d integer ==> xy = z^2 and since gdc(x,y) = 1 ==> x = m^2, y = n^2, where m,n are integers.

Therefore
a = m^4 + n^4, b = m^4 - n^4, c = 2m^2*n^2, d = mn^2*(m+n)

2. c = x^2 - y^2, b = 2xy

==>

x^2 - y^2 = 2d^2 / (2d - y(x-y)) ==>

2d = x^2 - y^2 +- ((x-y)sqrt(x^2-y^2)) and since d is positive,

2d = x^2 - y^2 + ((x-y)sqrt(x^2-y^2))

d integer ==> x^2 - y^2 = z^2 ==> x = m^2 + n^2, y = m^2 - n^2 or 2mn

2.1 : x = m^2 + n^2, y = m^2 - n^2

==>

a = (m^2+n^2)^2 + (m^2-n^2)^2

b = 2(m^2+n^2)(m^2-n^2)

c = (m^2+n^2)^2 - (m^2-n^2)^2

d = 2mn^2*(m+n)

2.2 : x = m^2 + n^2, y = 2mn

==>

a = (m^2+n^2)^2 + (m^2-n^2)^2

b = 4mn(m^2+n^2)

c = (m^2-n^2)^2

d = m(m+n)(m-n)^2

GOLDEN ISOSCELES TRIANGLE

Let ABC be an isosceles triangle with AB = AC and A'B'C'D' the inscribed square in ABC based on AC (A' on AB, B' on BC, C' on CA near C and D' on CA near A). Let y be the side of the square inscribed in BA'C' based on A'B' and z the side of the square inscribed in the right triangle D'A'A based on AD'.

If y = z, find the angles of the triangle.

Solution:

Let x be the side of the square A'B'C'D' and h the altitude BB*.

we have:

x = bh / (b+h)

y = bh^2 / (b+h)^2 = x^2 / b

z = xAD' / (x + AD') = x^2cotA /(x + xcotA) = xcotA / (1 + cotA)

y = z ==> x / b = cotA / (1 + cotA) ==>

(h/b) / (1 + (h/b)) = cotA / (1 + cotA) ==>

(h/c) / (1 + (h/c)) = cotA / (1 + cotA) ==>

sinA / (1 + sinA) = cotA / (1 + cotA) ==>

sinA = cotA ==> (cosA)^2 + cosA - 1 = 0 ==>

cosA = (-1 + sqrt(5))/2 (since 0 < A < 180) ==> A = Αrccos(1/φ) and B = C = 90 - (A/2)

Note: B* divides AC in golden ratio.

Addendum (13 April):

There is one more solution for A > 90 d.

See Hyacinthos Message 19985

REGULAR POLYGON PROBEM

Let A1A2A3... An be a regular n-gon. The perpendicular
to A1A2 at A2 intersects A3A4 at K. The parallel through
A3 to KA1 intersects A1A2 at M.

For which n's the point M is the midpoint of A1A2?

APH, Hyacinthos message 19937

Trigonometric Solution:

Let: A1A2 = A2A3 =... = 1, angles(A1A2A3) = (A2A3A4) = ... = ω

Denote: angle(KA1A2) = (A3MA2) = θ, angle(MA3A2) = φ, KA2 := d

We have:

In the right triangle A2KA1: tanθ = d

In the triangle KA2A3: d/sinω = 1/sin(270-2ω) ==> d / sinω = - 1/cos2ω

In the triangle MA2A3: (1/2)/sinφ = 1/sinθ ==> 2sinφ = sinθ
and since φ + θ = 180 - ω, ==> 2sin(ω+θ) = sinθ

So we have the system of two equations:

tanθ = - sinω/cos2ω

2sin(ω+θ) = sinθ

==>

2cos2ω - 2cosω + 1 = 0 ==> 4(cosω)^2 - 2cosω - 1 = 0 ==>

cosω = (1 +- sqrt5) / 4 = cos36 or cos108

We have:
180 > ω = 180(n-2)/n >= 60 since n >= 3 ==> ω = 108 d. and n = 5.

Other solutions by Nikos Dergiades, Hyacinthos messages
19939,19946

INRADIUS 4 (Golden Ratio)

Let ABC be a triangle and Aa a point on BC between B,C.

Denote:

h_a : the A-altitude of ABC

r_a : the A-exradius of ABC

r_ab, r_ac : the A-exradii of AAaB and AAaC.

We have:

h_a = 2r_ab.r_ac / (r_a - (r_ab + r_ac)) (1)

[See HERE]

Now, assume that r_ab = r_ac and r_a = 2h_a.

Denote r_ab / h_a := x

(1) ==>

1 = 2x^2 / (2 - 2x) ==> x^2 + x - 1 = 0

==>

0 < x = (- 1 + sqrt5)/2 = 1/φ, where φ is the "golden number" φ = (1 + sqrt5)/2.

Golden Section:

Let Iab, Iac be the A-excenters of AAbB, AAcC, resp., and A* the intersection of the line IabIac and the line of AA' (of the A-altitude).

A' divides ΑA* in golden ratio.

Addendum:

If the triangle ABC is isosceles AB = AC, then the foot A' of the A-altitude coincides with the point of contact of the A-excircle and BC:

CONSTRUCTION:

Let (A),(B) be two externally tangent circles at C with radius of (A) = 4.radius of (B). The line ACB intersects again the circle (B) at D. A tangent from D to circle (A) intersects the common internal tangent of (A),(B) at E. The bisector of the right angle ECA intersects EA at F. Let G be the orthogonal projection of F on AB. C divides GD in Golden Ratio.