HYACINTHOS 25820

 [Antreas P. Hatzipolakis]:

Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.

    Denote:

    Na, Nb, Nc = the NPC centers of PB'C', PC'A', PA'B', resp

    ABC, NaNbNc are parallelogic.

    The parallelogic center (ABC, NaNbNc) lies on the circumcircle.

    Which are the parallelogic centers in terms of P?

 

[Angel Montesdeoca]:
   

*** P=(u:v:w), barycentric coordinates.

The parallelogic center (ABC, NaNbNc) lies on the circumcircle:
U = (a^2 (b^4 w (-u+w)+(a^2-c^2) v (c^2 (u-v)+a^2 w)+b^2 (a^2 (u-v-w) w+c^2 (u^2+2 v w-u (v+w)))) : ... : ...).

The parallelogic center ( NaNbNc, ABC) is:
V = (a^2 (c^2 (a^2-c^2) v-b^4 w+b^2 (4 c^2 u+a^2 w)) : ... : ...).

*** When P moves on a line passing through the circumcenter, the point U remains fixed.

More details and figures in

http://amontes.webs.ull.es/ otrashtm/HGT2017.htm#HG180417

Angel Montesdeoca