HYACINTHOS 23914

 [Antreas P. Hatzipolakis]:

Let ABC be a triangle and A'B'C', A"B"C" the cevian, pedal triangles of G, resp.

Denote:

Ma, Mb, Mc = the midpoints of AA', BB', CC', resp.

Pa, Pb, Pc = same points on the Euler lines of A"MbMc, B"McMa, C"MaMb, resp.

Are the triangles ABC, PaPbPc orthologic for all Pa,Pb,Pc ?

 

And which are the loci of the orthologic centers as Pa,Pb,Pc move on the Euler lines being same?

Is the locus of the orthologic center (PaPbPc, ABC) the Euler line?

[Angel Montesdeoca]:

The locus of the orthologic centers (ABC, PaPbPc) is the circumconic isogonal conjugate of line passing through the centers X(187) = inverse-in-circumcircle of X(6) and  X(5054) = inverse-in orthocentroidal-circle of midpoint of  X(2) AND X(5).

The locus of the orthologic center ( PaPbPc, ABC) is the Euler line.P, Pa, Pb, Pc = same points on the Euler lines of  ABC, A"MbMc, B"McMa, C"MaMb, resp.

Let V be the orthologic center of PaPbPc with respect to ABC.

If OP:PH=t, then  OV:VH=(3+4t):(9+8t).
Pairs {P ,V}:   {X(3), X(140)}, {X(4), X(2)}, {X(5), X(10124)}, {X(20), X(549)},

If P=X(2), then V=( 8a^4-13a^2 (b^2+c^2)+5 (b^2-c^2)^2:...:...) is the  midpoint of X(i) and X(j) for these {i,j}:  {2,5054}, {3,3545}, {3524,5055}.

If P=X(140), then V=(  22 a^4-35 a^2 (b^2+c^2)+13 (b^2-c^2)^2:...:...) is the  midpoint of X(i) and X(j) for these {i,j}:  {140,10124}, {376,3861}, {547,3530}, {549,3628}, {3860,8703}.

if P= 8 X(3) +  3 X(4) = X(3525) then  V= X(3525) and Pa, Pb, Pc are the X(3525) of triangles A"MbMc, B"McMa, C"MaMb.

if P = 8 X(3) - 9 X(4) = (17 a^4 - 8 a^2 (b^2 + c^2)- 9 (b^2 - c^2)^2:...:...) the triangles  ABC and PaPbPc are orthologic and perspective, with perspector V=X(4).

Angel  Montesdeoca