HYACINTHOS 25830

 [Antreas P. Hatzipolakis]:

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

Denote:

T1, T2, T3 = the pedal triangles of A, B, C wrt triangle A'B'C'.

Na, Nb, Nc = the NPC centers of T1, T2, T3, resp.

4. The circumcenter of NaNbNc lies on the OI line. Point? 

 

[Angel Montesdeoca]:

****   (r+3 R) X(1) + (R-r) X(3)

(a (a^5 (b+c)-a^4 (b^2-4 b c+c^2)-2 a^3 (b^3+b^2 c+b c^2+c^3)+2 a^2 (b^4-3 b^3 c+2 b^2 c^2-3 b c^3+c^4)+a (b-c)^2 (b+c)^3-(b-c)^4 (b+c)^2):  ... : ...),

Midpoint of X(i) and X(j) for these {i,j}:  {5,12675}, {942,1385}, {1125,12005}, {1483,5836}, {3881,6684}, {5045,9940}, {5083,6713}.

On lines:  {1,3}, {5,3742 }, {72,6878}, {119,9947}, {140,518}, {143,9037}, {210,3526}, {355,5439 }, {381,12680}, {496,10391}, {551,5884},{575,9004}, ...

with (6,9,13)- search numbers (2.67317655665387, 2.50188834925857, 0.674814137042048).

Angel Montesdeoca