Dear Antreas,

[APH]: Let ABC be a triangle, P a point, A', B', C' the orth.

projections of H on AP, BP, CP, resp. and Ea, Eb, Ec

the midpoints of AH, BH, CH (ie EaEbEc = the Euler

triangle of ABC)

Which is the locus of P such that A'B'C', EaEbEc

are perspective? (Locus of the perspectors?)

I think that I, N are points of the locus.

*** Yes, the locus the quintic

cyclic sum a^2(S_{CA}+S_{AB}+2S_{BC})v^2w^2(S_Bv - S_Cw)

+ cyclic uvw S_A(S_B-S_C)(b^2c^2u^2- a^2S_Avw) = 0.

It also contains H (trivially), X_{80}, and X_{1263}. Note that X(80)

and X(1263) are respectively the reflection conjugates of I and N

respectively.

I have verified that the quintic is indeed invariant reflection

conjugation. [The reflection conjugate of a point P is the common

point of the reflections of the circles PBC, PCA, PAB in the sidelines

BC, CA, AB respectively].

Best regards

Sincerely

Paul