HYACINTHOS 26100

[Antreas P. Hatzipolakis]:

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

Denote:

A"B"C" = the reflection triangle of ABC
(A", B", C" = the reflecctions of A, B, C in BC, CA, AB, resp.)

La, Lb, Lc = the Euler lines of A"B'C', B"C'A', C"A'B', resp.

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

ABC, A*B*C* are parallelogic.
The parallelogic center (ABC, A*B*C*) is the N.
The other one (A*B*C*, ABC) ?


[Angel Montesdeoca]:

*** The parallelogic center (A*B*C*, ABC)  is V=X(5)+X(3060)

V = (a^2 (2 a^6 (b^2+c^2)-6 a^4 (b^4+b^2 c^2+c^4)+a^2 (6 b^6-5 b^4 c^2-5 b^2 c^4+6 c^6)-(b^2-c^2)^2 (2 b^4-5 b^2 c^2+2 c^4)) : ... : ...),

V is the midpoint of X(i) and X(j) for these {i,j}: {5,3060}, {568,3845}, {3917,10263}, {5446,5943}.

V is  reflection of X(i) in X(j) for these {i,j}:   {140, 5943}, {547, 13364}, {3917, 3628}, {5447, 10219}, {5891, 11737}, {5943, 10095}, {12100, 13363}.

V lies on lines X(i)X(j) for these {i,j}:  {4, 13321}, {5, 3060}, {26, 3527}, {30, 51}, {52, 3850}, {140, 5446}, {143, 546}, {185, 12102}, {373, 10124}, {381, 11002}, {389, 3853}, {511, 547}, {548, 5462}, {549, 5640}, {568, 3845}, {1112, 11801}, {1154, 5066}, {1216, 12812}, {1658, 10982}, {1994, 7545}, {3567, 3627}, {3628, 3917}, {3856, 5876}, {3858, 5889}, {3859, 5907}, {3861, 6102}, {5447, 10219}, {5562, 12811}, {5891, 11737}, {6000, 12101}, {6030, 13353}, {7530, 9777}, {8254, 13383}, {9729, 12002}, {9969, 10272}, {11451, 11539}, {11793, 13421}, {12006, 12103}, {12100, 13363}, {12834, 13339}.

with (6-9-13)-search number (-0.917622942719411, -1.19166978867973, 4.88918492455623).


Angel Montesdeoca