Let ABC be a triangle and A'B'C' the cevian triangle of P = I
Denote:
Ab, Ac = the orthogonal projections of A on BB',CC', resp.
Bc, Ba = the orthogonal projections of B on CC',AA', resp.
Ca, Cb = the orthogonal projections of C on AA',BB', resp.
Abc = the orthogonal projection of Ab on CC'
Acb = the orthogonal projection of Ac on BB'
Bca = the orthogonal projection of Bc on AA'
Bac = the orthogonal projection of Ba on CC'
Cab = the orthogonal projection of Ca on BB'
Cba = the orthogonal projection of Cb on AA'
For P = I, we have Bca = Cba =: A*, Cab = Acb =: B* and Abc = Bac =: C*
The Euler lines of AB*C*, BC*A*, CA*B* are concurrent.
Which is the locus of P such that the Euler lines of AAbcAcb, BBcaBac, CCabCba are concurrent?
Note: The Euler Lines of AAbAc, BBcBa, CCaCb are concurrent at Feuerbach point.
(APH, Hyacinthos)
Orthic Triangle Version:
Let ABC be a triangle and A'B'C' the cevian/pedal triangle of P = H (orthic triangle)
Denote:
A'b, A'c = the orthogonal projections of A' on BB',CC', resp.
B'c, B'a = the orthogonal projections of B' on CC',AA', resp.
C'a, C'b = the orthogonal projections of C' on AA',BB', resp.
A'bc = the orthogonal projection of A'b on CC'
A'cb = the orthogonal projection of A'c on BB'
B'ca = the orthogonal projection of B'c on AA'
B'ac = the orthogonal projection of B'a on CC'
C'ab = the orthogonal projection of C'a on BB'
C'ba = the orthogonal projection of C'b on AA'
For P = H, we have B'ca = C'ba =: A*, C'ab = A'cb =: B* and A'bc = B'ac =: C*
The Euler lines of A'B*C*, B'C*A*, C'A*B* are concurrent.
Let A'B'C' be the cevian (or pedal) triangle of P. Which is the locus of P such that the Euler lines of A'A'bcA'cb, B'B'caB'ac, C'C'abC'ba are concurrent?
Summary:
Let ABC be a triangle, P a point and A'B'C' the cevian triangle of P.
1. Denote:
Ab, Ac = the orthogonal projections of A on BB',CC', resp.
Bc, Ba = the orthogonal projections of B on CC',AA', resp.
Ca, Cb = the orthogonal projections of C on AA',BB', resp.
Abc = the orthogonal projection of Ab on CC'
Acb = the orthogonal projection of Ac on BB'
Bca = the orthogonal projection of Bc on AA'
Bac = the orthogonal projection of Ba on CC'
Cab = the orthogonal projection of Ca on BB'
Cba = the orthogonal projection of Cb on AA'
Which is the locus of P such that the Euler lines of AAbcAcb, BBcaBac, CCabCba are concurrent?
2. Denote:
A'b, A'c = the orthogonal projections of A' on BB',CC', resp.
B'c, B'a = the orthogonal projections of B' on CC',AA', resp.
C'a, C'b = the orthogonal projections of C' on AA',BB', resp.
A'bc = the orthogonal projection of A'b on CC'
A'cb = the orthogonal projection of A'c on BB'
B'ca = the orthogonal projection of B'c on AA'
B'ac = the orthogonal projection of B'a on CC'
C'ab = the orthogonal projection of C'a on BB'
C'ba = the orthogonal projection of C'b on AA'
Which is the locus of P such that the Euler lines of A'A'bcA'cb, B'B'caB'ac, C'C'abC'ba are concurrent?
Antreas P. Hatzipolakis, 6 March 2013
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