HYACINTHOS 26773

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle, A'B'C' the cevian triangle of H and A"B"C" the cevian triangle of O.
Let Ma, Mb, Mc be the midpoints of AA", BB", CC", resp. and M1, M2, M3 the midpoints of A'Ma, B'Mb, C'Mc, resp.

A'B'C', M1M2M3 are orthologic.

 

[Peter Moses]:

Hi Antreas,

(A'B'C',  M1M2M3): 

P = a^2 (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2-a^4 b^2 c^2-a^2 b^4 c^2+b^6 c^2-3 a^4 c^4-a^2 b^2 c^4+3 a^2 c^6+b^2 c^6-c^8) (a^10 b^2-4 a^8 b^4+6 a^6 b^6-4 a^4 b^8+a^2 b^10+a^10 c^2-2 a^8 b^2 c^2+3 a^4 b^6 c^2-3 a^2 b^8 c^2+b^10 c^2-4 a^8 c^4+2 a^4 b^4 c^4+2 a^2 b^6 c^4-4 b^8 c^4+6 a^6 c^6+3 a^4 b^2 c^6+2 a^2 b^4 c^6+6 b^6 c^6-4 a^4 c^8-3 a^2 b^2 c^8-4 b^4 c^8+a^2 c^10+b^2 c^10):: 

on line {5,389}.
P is on the Jerabek hyperbola of the orthic triangle.

This leads us to the conclusion that P is Q of the orthic triangle, where 

Q = (b+c) (a^3+b^3+a b c-a c^2-b c^2) (a^3-a b^2+a b c-b^2 c+c^3):: 

on lines {{3,10},{4,12930},{6,1826},{12,73},{42,10950},{64,1869},{65,1867},{66,11391},{68,10526},{69,313},{71,594},{72,1089},{80,5247},{265,2779},{333,5086},{1146,2333},{1220,1798},{1243,7686},{1245,1834},{3185,10454},{3519,12936},{4267,10572},{5136,14529},{5504,12890}}.
{X(10570),X(13478)}-harmonic conjugate of X(2217).
isogonal conjugate of X(4225).
isogonal of the anticomplement X(3142).
X(181)-cross conjugate of X(37).
X(i)-isoconjugate of X(j) for these (i,j): {{1, 4225}, {21, 10571}, {58, 3869}, {81, 573}, {86, 3185}, {662, 6589}, {1333, 4417}, {1444, 3192}}.
crosspoint of X(2995) and X(13478).
trilinear pole of line {647, 4024}.
crosssum of X(573) and X(3185).
barycentric product X(i)X(j) for these {i,j}: {{10, 13478}, {37, 2995}, {226, 10570}, {321, 2217}}.
barycentric quotient X(i)/X(j) for these {i,j}: {{6, 4225}, {10, 4417}, {37, 3869}, {42, 573}, {213, 3185}, {512, 6589}, {1400, 10571}, {2217, 81}, {2333, 3192}, {2995, 274}, {10570, 333}, {13478, 86}}.
on the Jerabek hyperbola.
on the cubic K901.

(M1M2M3, A'B'C'): X(389).

Best regards,
Peter Moses.