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 X(1317).

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 10485)

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

Addition (31 Aug, 2026)

Let ABC be a triangle and A'B'C' the pedal triangle of I.

Denote

A* = the orthogonal projection of A' on AI
Ab, Ac = the orthogonal projections of A* on BI,CI, resp.

La = the Euler line of A*AbAc
Similarly Lb, Lc.

1. La, Lb, Lc are concurrent. Point?

2.The parallels to La, Lb, Lc through A, B, C, resp. are concurrent. Point?

3.The parallels to La, Lb, Lc through A', B', C', resp. are concurrent.
Point = the antipode of Feuerbach point X(11) in the incircle = X(1317)
Euclid 10155