### Euler line of orthic triangle

Hyacinthos #24787

[Antreas Hatzipolakis]:

Let ABC be a triange and P a point.

Denote:

A', B', C' = the reflections of P in BC, CA, AB, resp.

Ab, Ac = the orthogonal projections of A' on AC, AB, resp.

A2, A3 = the orthogonal projections of P on A'Ab, A'Ac, resp.

Bc, Ba = the orthogonal projections of B' on BA, BC, resp.
B3, B1 = the orthogonal projections of P on B'Bc, B'Ba, resp.

Ca, Cb = the orthogonal projections of C' on CB, CA, resp.

C1, C2 = the orthogonal projections of P on C'Ca, C'Cb, resp.

Which is the locus of P such that the perpendicular bisectors of A2A3, B3B1, C1C2 are concurrent?

[Angel Montesdeoca]:

The locus of P such that the perpendicular bisectors of A2A3, B3B1, C1C2 are concurrent is the Euler line and the point Q of concurrence lies on Euler line of orthic triangle.

Pairs {P,Q}: {2,51}, {3,5}, {4,52}, {5,143}, {20,5562}, {22,343}, {140,10095}, {376,5891}, {2071,1568}, {7512,1209}

When P traverses the Euler line the envelope of lines PQ is the parabola of focus Tixier point X(476) (the reflection of X(110) in the Euler line) and directrix X(110)X(351) (X(110)= focus Kiepert parabola, X(351) = center of the Parry circle).

This parabola passes through the points :

X(3)= circumcenter,

X(143) = nine-point center of orthic triangle,

X(5640) = centroid of orthocentroidal triangle,

X(5663) = point of intersection of the Euler line of orthocentroidal triangle and the line at infinity,

X(5889) = orthocemter of circumorthic triangle

http://amontes.webs.ull.es/otrashtm/HGT2016.htm#HG091116

Angel Montesdeoca. Noviembre, 2016