[Antreas P. Hatzipolakis]:
Let ABC be a triangle and A'B'C' the pedal triangle of N.
Denote:
Ab, Ac = the orthogonal projections of A' on NB', NC', resp.
Bc, Ba = the orthogonal projections of B' on NC', NA' resp.
Ca, Cb = the orthogonal projections of C' on NA', NB', resp.
A2, A3 = the orthogonal projections of A' on BcBa, CaCb, resp.
B3, B1 = the orthogonal projections of B' on CaCb, AbAc, resp.
C1, C2 = the orthogonal projections of C' on AbAc, BcBa, resp.
Oa, Ob, Oc = the circumcenters of A'A2A3, B'B3B1, C'C1C2, resp.
1. ABC, OaObOc are homothetic.
2. ABC, OaObOc are orthologic.
Orthologic centers ?
[César Lozada]:
1) Homothetic center: X(373) = CENTROID OF THE PEDAL TRIANGLE OF THE CENTROID
2) Orthologic centers:
Z(A->Oa) = H
Z(Oa->A) = (cos(2*A)-2)*cos(B-C)-3*cos(A) : : (trilinears)
= a*((b^2+c^2)*a^6-(3*b^4-4*b^2* c^2+3*c^4)*a^4+(b^2+c^2)*(3*b^ 4-14*b^2*c^2+3*c^4)*a^2-(b^4- 4*b^2*c^2+c^4)*(b^2-c^2)^2) : : (trilinears)
= 3*X(2)+X(389) = X(4)-9*X(373)
= On lines: {2,389}, {3,5943}, {4,373}, {5,2883}, {20,11451}, {26,5092}, {51,631}, {52,3526}, {140,143}, {159,182}, {185,3090}, {549,5446}, {575,1147}, {578,10601}, {632,1216}, {970,6883}, {1092,5422}, {1181,11284}, {1199,3292}, {1498,11484}, {1656,5907}, {1995,10984}, {2818,3812}, {3060,10303}, {3066,11414}, {3357,5544}, {3517,5085}, {3523,5640}, {3524,9781}, {3525,3567}, {3530,10095}, {3533,5650}, {3545,11381}, {3628,10219}, {3851,10575}, {5020,6759}, {5054,10625}, {5056,10574}, {5067,5890}, {5070,5891}, {5071,6241}, {5651,7592}, {6101,11539}, {6102,10170}, {6643,9815}, {6723,9826}, {7395,11438}
= midpoint of X(i) and X(j) for these {i,j}: {3,10110}, {5,9729}, {140,5462}, {143,5447}, {182,9822}, {3530,10095}, {5892,6688}, {6723,9826}
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (3,5943,10110), (4,11465,373), (5,5892,9729), (52,3526,3819), (140,143,5447), (182,6642,10282), (632,5946,1216), (1656,9730,5907), (3525,3567,3917), (3533,11412,5650), (5447,5462,143), (6688,9729,5)
= [ 2.726838614247667, 2.07251985890066, 0.947302142246541 ]
César Lozada
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