[Antreas P. Hatzipolakis]:
Let ABC be a triangle and A'B'C' the pedal triangle of H.
Let A", B", C" be three arbitrary points on AA', BB', CC', resp.
ABC, A"B"C" are orthologic.
Application:
Let ABC be a triangle and A'B'C' the pedal triangle of O.
Denote:
A"B"C" = the reflection triangle of ABC
(A", B", C" = the reflections of A, B, C in BC, CA, AB, resp.)
Ha, Hb, Hc = the orthocenters of A"B'C', B"C'A', C"A'B', resp.
ABC, HaHbHc are orthologic.
Which is the orthologic center (ABC, HaHbHc) ?
Let A", B", C" be three arbitrary points on AA', BB', CC', resp.
ABC, A"B"C" are orthologic.
Application:
Let ABC be a triangle and A'B'C' the pedal triangle of O.
Denote:
A"B"C" = the reflection triangle of ABC
(A", B", C" = the reflections of A, B, C in BC, CA, AB, resp.)
Ha, Hb, Hc = the orthocenters of A"B'C', B"C'A', C"A'B', resp.
ABC, HaHbHc are orthologic.
Which is the orthologic center (ABC, HaHbHc) ?
[Peter Moses]:
Hi Antreas,
a^2 (2 a^4-4 a^2 b^2+2 b^4+3 a^2 c^2+3 b^2 c^2-5 c^4) (2 a^4+3 a^2 b^2-5 b^4-4 a^2 c^2+3 b^2 c^2+2 c^4)::
on lines {{20,3519},{54,3357},{ 64,3518},{68,3529},{186,3532}, {265,3146},{3090,4846},{3091, 3521},{3426,10594},{3431,6241} ,{3531,11403},{3542,10293},{ 5365,11138},{5366,11139},{ 6000,11270},{6415,6447},{6416, 6448},{8717,11440},{10990, 11564},{11381,11738},{11744, 12250}}.
on Jerabek.
isogonal conjugate of X(1657).
X(12290)-cross conjugate of X(4).
X(i)-vertex conjugate of X(j) for these (i,j): {{3, 11270}, {4, 3532}}.
barycentric quotient X(6)/X(1657).
Searches: {18. 8472809179835384476826215201, 19. 4315365530680094417850224010,- 18. 5106827862089528708940234152}.
Best regards,
Peter Moses.
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