[Tran Quang Hung, rephrased]:
Let ABC be a triangle, P a point and A'B'C' the circumcevian triangle of P.
Denote:
A1A2A3, B1B2B3, C1C2C3 = the pedal triangles of A', B', C' wrt triangles PBC, PCA, PAB, resp.
La, Lb, Lc = the Euler lines of A1A2A3, B1B2B3, C1C2C3, resp.
Which is the locus of P such that La, Lb, Lc are concurrent?
[Angel Montesdeoca]:
*** The locus of P such that La, Lb, Lc are concurrent is a degree-14-circumcurve passing through the vertices of the excentral triangle, X(1), X(4), X(74).
For P=I the concurrent point is X(21)
For P=H the concurrent point is X(52)
For P=X(74) the concurrent point is the midpoint of X(3) and X(74):
(a^2 (2 a^8-3 a^6 (b^2+c^2)-3 a^4 (b^4-4 b^2 c^2+c^4)-(b^2-c^2)^2 (3 b^4+7 b^2 c^2+3 c^4)+a^2 (7 b^6-8 b^4 c^2-8 b^2 c^4+7 c^6)) : ... : ....),
with (6-9-13)-search numbers
(9.719762854243672961545 , 9.2927959862678684885096 ,-7.278854056698148140409).
Angel Montesdeoca
For P=H the concurrent point is X(52)
For P=X(74) the concurrent point is the midpoint of X(3) and X(74):
(a^2 (2 a^8-3 a^6 (b^2+c^2)-3 a^4 (b^4-4 b^2 c^2+c^4)-(b^2-c^2)^2 (3 b^4+7 b^2 c^2+3 c^4)+a^2 (7 b^6-8 b^4 c^2-8 b^2 c^4+7 c^6)) : ... : ....),
with (6-9-13)-search numbers
(9.719762854243672961545 , 9.2927959862678684885096 ,-7.278854056698148140409).
Angel Montesdeoca
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