Τετάρτη, 6 Μαρτίου 2013

CONCURRENT EULER LINES

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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