FEUERBACH POINT

Let ABC be a triangle and A'B'C', A"B"C" the cevian, pedal triangles of I, resp.

Denote:

Ab, Ac = the reflections of A' in BB', CC'

Bc, Ba = the reflections of B' in CC', AA'

Ca, Cb = the reflections of C' in AA', BB'

Ea, Eb, Ec = the Euler Lines of A'AbAc, B'BcBa, C'CaCb, resp. (concurrent at I)

1. Antipode of Feuerbach Point.

Denote:

A2,A3 = the orthogonal projections of A on Eb, Ec, resp.

B3,B1 = the orthogonal projections of B on Ec, Ea, resp.

C1,C2 = the orthogonal projections of C on Ea, Eb, resp.

The Euler lines L1,L2,L3 of AA2A3, BB3B1, CC1C2, resp. are concurrent at the antipode of the Feuerbach point.

Note:

Denote:

12, 13 = the orthogonal projections of A on BB', CC', resp.

23, 21 = the orthogonal projections of B on CC', AA', resp.

31, 32 = the orthogonal projections of C on AA', BB', resp.

The Euler lines of A1213, B2321, C3132 are concurrent at Feuerbach point

(APH, Hyacinthos)

2. Antipode of Feuerbach Point.

Denote:

La = the parallel to Ea through A"

Lb = the parallel to Eb through B"

Lc = the parallel to Ec through C"

The La,Lb,Lc are concurrent at the antipode of the Feuerbach point.

Antreas P. Hatzipolakis, 25 Febr. 2013