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