### Concurrent Euler Lines

Hyacinthos #24829

Let ABC be a triangle and HaHbHc the pedal riangle of H.

Denote:

A*,B*,C* = points on AHa, BHb, CHc, resp. such that A*A / A*Ha = B*B/B*Hb = C*C/C*Hc = t.

A2, A3 = the orthogonal projections of A* on AC, AB, resp.
Smilarly B3, B1 and C1, C2.

The Euler lines of

1. AA2A3, BB3B1, CC1C2

2. A*A2A3, B*B3B1, C*C1C2

are concurrent.

1. The loci of the points of concurrence Q of Euler lines of AA2A3, BB3B1, CC1C2 as t varies is the line X(6)X(74)

2. The loci of the points of concurrence Q* of Euler lines of A*A2A3, B*B3B1, C*C1C2 as t varies isthe line The line X(6)X(24).

The envelope of line QQ* is the parabola tangent to lines X(24)X(74), X(6)X(24) and X(6)X(74) at X(5621).

With these data the parabola can be constructed (see for example, Paris Pamfilos.- A Gallery of Conics by Five Elements):

§13.2. Parabola by 1 tangent-at, 2 tangents (1P4T_{1})_{1}

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

Angel Montesdeoca. Nov. 2016