HYACINTHOS 26077

[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 BH, CH, resp.

La = the Euler line of A'AbAc. Similarly Lb, Lc.

A*B*C* = the triangle bounded by La,Lb,Lc.

ABC, A*B*C* are parallelogic.

 

[Angel Montesdeoca]:

=== The parallelogic center (ABC, A*B*C*) is:

W = (a^2 (-a^8+a^6 (b^2+c^2)+a^4 (3 b^4-11 b^2 c^2+3 c^4)+a^2 (-5 b^6+7 b^4 c^2+7 b^2 c^4-5 c^6)+(b^2-c^2)^2 (2 b^4+7 b^2 c^2+2 c^4)) : ... : ....),

with with (6-9-13)-search number (19.2144490890057, 19.2306290337570, -18.5410551979269).

W is reflection of X(i) in X(j) for these {i,j}: {110,2071}, {146,1568}, {2070,12041}, {3448,13399}.

=== The parallelogic center ( A*B*C*,ABC) is:

Z = (a^2 (a^12 (b^2+c^2)
     -2 a^10 (2 b^4+3 b^2 c^2+2 c^4)
     +a^8 (5 b^6+6 b^4 c^2+6 b^2 c^4+5 c^6)
     +a^6 b^2 c^2 (b^4-20 b^2 c^2+c^4)
     +a^4 (-5 b^10+6 b^8 c^2+3 b^6 c^4+3 b^4 c^6+6 b^2 c^8-5 c^10)
     +a^2 (b^2-c^2)^2 (4 b^8-7 b^6 c^2+12 b^4 c^4-7 b^2 c^6+4 c^8)
     -(b^2-c^2)^4 (b^6-3 b^4 c^2-3 b^2 c^4+c^6) ) : ... : ....),

with with (6-9-13)-search number (-2.96800723437968, -3.38490820591826, 7.35391196341071).

Z is the midpoint of X(i) and X(j) for these {i,j}:  {143,11558}, {5446,11563}.

Z is the reflection of X(13376) in X(10110).

Angel Montesdeoca