[Antreas P. Hatzipolakis]:
Let ABC be a triangle.
Denote:
A' = point on AH such that A'A/A'H = t
B' = point on BH such that B'A/B'H = t
C' = point on CH such that C'A/C'H = t
Ab, Ac = the orthogonal projections of A' on AB, AC, resp.
Bc, Ba = the orthogonal projections of B' on BC, BA, resp.
Ca, Cb =the orthogonal projections of C' on CA,CB, resp.
1. The Euler lines of A'AbAc, B'BcBa, C'CaCb are concurrent
2. The Euler lines of AAbAc, BBcBa, CCaCb are concurrent.
Denote:
A' = point on AH such that A'A/A'H = t
B' = point on BH such that B'A/B'H = t
C' = point on CH such that C'A/C'H = t
Ab, Ac = the orthogonal projections of A' on AB, AC, resp.
Bc, Ba = the orthogonal projections of B' on BC, BA, resp.
Ca, Cb =the orthogonal projections of C' on CA,CB, resp.
1. The Euler lines of A'AbAc, B'BcBa, C'CaCb are concurrent
2. The Euler lines of AAbAc, BBcBa, CCaCb are concurrent.
[Angel Montesdeoca]:
*** The Euler lines of A'AbAc, B'BcBa, C'CaCb concur at W. The locus of W is the line X(4)X(54).
*** The Euler lines of AAbAc, BBcBa, CCaCb concur at Z. The locus of Z is the line X(4)X(74).
The lines WZ are parallel, and meet the line at infinity at T.
T = (a^2 (a^12 (b^2+c^2)
-2 a^10 (2 b^4+b^2 c^2+2 c^4)
+a^8 (5 b^6+2 b^4 c^2+2 b^2 c^4+5 c^6)
+a^6 (-5 b^6 c^2+4 b^4 c^4-5 b^2 c^6)
-a^4 (b^2-c^2)^2 (5 b^6+2 b^4 c^2+2 b^2 c^4+5 c^6)
+a^2 (b^2-c^2)^2 (4 b^8+3 b^6 c^2+3 b^2 c^6+4 c^8)
-(b^2-c^2)^4 (b^2+c^2)^3) : ... : ...),
with (6-9-13)-search numbers (8.85525918256804, 0.909856465095148, -4.71694332932804)
T is the infinity point of lines:
{3, 8157}, {4, 7730}, {52, 265}, {54, 74}, {110, 5562}, {113, 1209}, {125, 389}, {146, 2888}, {195, 2935}, {399, 2917}, {973, 1112}, {1205, 6776}, {1216, 1511}, {1498, 5898}, {1539, 6153}, {3448, 5889}, {5972, 7542}, {6276, 7726}, {6277, 7725}, {6288, 7728}, {6689, 6699}, {6723, 9826}, {7356, 7727}, {7978, 7979}, {8994, 8995}, {9904, 9905}, {9984, 9985}, {10065, 10066}, {10081, 10082}.
Angel Montesdeoca
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