X(1112) y un centro del triángulo relacionado
Let ABC be a triangle and A'B'C' the orthic triangle (pedal triangle of H).
Ab, Ac = the orthogonal projections of A' on BH, CH, resp.
Bc, Ba = the orthogonal projections of B' on CH, AH, resp.
Ca, Cb = the orthogonal projections of C' on AH, BH, resp.
The Euler lines La, Lb, Lc of triangles A'AbAc, B'BcBa, C'CaCb concur at X(1112).
X(1112) is the center of the hyperbola that passes through the vertices of the cevian triangles of X(4) and X(648), and also through the centers X(i) for i = 4, 113, 155, 193, 1162, 1163, 1829, 1839, 1843, 1858, 1986, 2574, 2575, 2904, 2905, 2906, 2907, 2914, 3574, 5095, 5895, 6152, 10294, 11560, 11817, 13202, 13420.
Let Oa, Ha be the circumcenter and the orthocenter of triangle A'AbAc, resp.
Let Ob, Hb be the circumcenter and the orthocenter of triangle B'BcBa, resp.
Let Oc, Hc be the circumcenter and the orthocenter of triangle C'CaCb, resp.
Let Pa, Pb, Pc be points on La, Lb, Lc, resp., such that OaPa/PaHa = ObPb/PbHb = OcPc/PcHc.
If Pa=X(1112) let da be the line PbPc, and define the lines db and dc cyclically.
The lines da, db, dc concur at the point W with homogeneous barycentric coordinates:
W = (2 a^12
-2 a^10 (b^2+c^2)
+a^8 (-3 b^4+8 b^2 c^2-3 c^4)
+2 a^6 (b^2-c^2)^2 (b^2+c^2)
+2 a^4 (b^2-c^2)^2 (b^4-b^2 c^2+c^4)
-(b^2-c^2)^4 (b^2+c^2)^2 : ... : ....).
W = (r(r+4R)-s^2) X(4) - ((r+2 R)^2-s^2) X(110)
W = (r^2+4 r R+6 R^2-s^2) X(25) - ((r+2 R)^2-s^2) X(125)
W is the midpoint of X(i) and X(j) for these {i,j}: {4,12140}, {3575,12133}, {7553,12358}, {7687,13419}.
W lies on lines X(i)X(j) for these {i,j}: {4, 110}, {24, 6699}, {25, 125}, {30, 13416}, {67, 7716}, {74, 7487}, {235, 7687}, {265, 1598}, {427, 5972}, {428, 542}, {468, 6723}, {541, 7576}, {1503, 11746}, {1511, 1595}, {1594, 12900}, {1596, 10113}, {1597, 12121}, {1843, 13417}, {2777, 3575}, {3448, 6995}, {3867, 6593}, {5064, 5642}, {5095, 12167}, {5663, 6756}, {7540, 7723}, {7553, 12358}, {7713, 13211}, {7714, 9140}, {7715, 10264}, {7718, 7984}, {11363, 11735}, {12173, 13202}.
http://amontes.webs.ull.es/otrashtm/HGT2017.htm#HG050617
Angel Montesdeoca. Junio, 2017