HYACINTHOS 26050

 [Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A'B'C' the pedal triangle of O.

Denote:

A", B", C" points on OA', OB', OC', resp. such that:

A"O/A"A' = B"O/B"B' = C"O/C"C' = t

Ab, Ac = the orthogonal projections of A" on OB, OC, resp.

Bc, Ba = the orthogonal projections of B" on OC, OA, resp.

Ca, Cb = the orthogonal projections of C" on OA, OB, resp.

A*B*C* = the triangle bounded by AbAc, BcBa, CaCb.

ABC, A*B*C* are homothetic.

Which is the locus of the homothetic center as t varies?

[César Lozada]:

Locus = Line through ETC’s:

{ 3, 49, 155, 184, 185, 283, 394, 1092, 1147, 1181, 1204, 1216, 1437, 1790, 1800, 1801, 1819, 3167, 3292, 3796, 3917, 5406, 5407, 5408, 5409, 5447, 5562, 7689, 8913, 9703, 9704, 9720, 9908, 10132, 10133, 10605, 10670, 10674, 10984}

= polar trilinear of X(4558)

= isogonal of the rectangular hyperbola CyclicSum[b*c*(SB-SC)*SB*SC*v* w)]=0, with center=X(136), perspector=X(2501) and passing through ETC’s: 4, 93, 225, 254, 264, 393, 847, 1093, 1105, 1179, 1217, 1300, 1826, 6344, 6526, 6531, 8737, 8738, 8741, 8742, 8801, 8884

 

If the homothetic center is Z(t), then

OZ(t) = t*(7*R^2-2*SW)/(2*(2*R^2+(3*R^ 2-SW)*t))*X(3)X(49)

 

César Lozada