HOMOTHETIC TRIANGLES and EULER LINES

Theorem 1.

Let ABC be an equilateral triangle and P a point. The Euler lines of the triangles PBC,PCA,PAB are concurent.Denote the point of concurrence with P*.

Reference:

APH, Hyacinthos #21592

Locus Problems:

As P moves on a line or on a circle [special case the incircle of ABC] which is the locus of P*?

Application:

Let ABC be a triangle and A',B',C' the apices of the equilateral triangles erected out/inwardly ABC, P a point and Pa, Pb, Pc the respective points of concurrences of the Euler lines wrt equil. triangles A'BC,B'CA,C'AB.

Which is the locus of P such that ABC, PaPbPc are perspective or orthologic?

Theorem 2 (generalization of Th. 1).

Let ABC, A'B'C' be two homothetic equilateral triangles. The Euler lines of the triangles AB'C', BC'A', CA'B' (and of the triangles A'BC, B'CA, C'AB) are concurrent.

Application to Morley Configuration with homothetic equilateral triangles:

1st Morley Triangle

2nd Morley Triangle

3rd Morley Triangle

Roussel Triangle

Theorem 3.

Let ABC, A'B'C' be two dilated triangles with scale factor 1. The Euler lines of AB'C', BC'A', CA'B' (and of the triangles A'BC, B'CA, C'AB) are concurrent.

Application:

Let ABC be a triangle and (O1),(O2),(O3) the reflections of the circumcircle (O) in BC,CA,AB, resp. Denote:

Ab = the second intersection of (O2) and the reflection of AN in the bisector of the angle HAC.

Ac = the second intersection of (O3) and the reflection of AN in the bisector of the angle HAB.

Similarly (cyclically) Bc, Ba and Ca, Cb

Oa, Ob, Oc = the circumcenters of the triangles ABcCb, BCaAc, CAbBa

The triangles ABC, OaObOc are dilated triangles with scalar facror 1. The Euler lines of AObOc, BOcOa, COaOb are concurrent and also the Euler lines of OaBC, ObCA, OcAB.

Note: The circumradii of the triangles ABcCb, BCaAc, CAbBa are equal.

Antreas P. Hatzipolakis, 20 April 2014