Τετάρτη 23 Οκτωβρίου 2019

HYACINTHOS 25389

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and OaObOc the pedal triangle of O (medial triangle)

Denote:

A', B', C' = the reflections of I in BC, CA, AB, resp.
Ma, Mb, Mc = the midpoints of AA', BB', CC', resp.
Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.

The circumcircles of OaMaNa, ObMbNb, OcMcNc are coaxial.

 

[Angel Montesdeoca]:

 

 

**** Intercept of line of centers and the common radical axis is
 (r^2-4 r R+s^2) X(1) - (r^2-12 r R+s^2)  X(106),
 of barycentric coordinates:

(a (a^3 (b-c)^2+a^2 b c (b+c)-a (b^4-5 b^3 c+12 b^2 c^2-5 b c^3+c^4)-b c (b^3-2 b^2 c-2 b c^2+c^3)) : ... : ... ),

with (6-9-13)-search numbers (2.30850549102976,  -0.464378057383825,  2.89669291039022).


**** X(2802) = intercept of line of centers and the line at infinity

Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically. X(2802) = X(519)-of-IaIbIc. (Randy Hutson, February 10, 2016).   X(519) = intercept of Nagel line and the line at infinity.

Angel Montesdeoca

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