## Παρασκευή, 25 Νοεμβρίου 2011

### 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

## Δευτέρα, 21 Νοεμβρίου 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

## Σάββατο, 19 Νοεμβρίου 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