## Τετάρτη, 22 Φεβρουαρίου 2012

### Concurrent Euler Lines (generalization)

Problem 1:

Let L1,L2 be two lines intersected at A, and P a point. To draw line L intersecting L1,L2 at Ab,Ac, resp. such that:

P be

- the circumcenter of AAbAc

Ab, Ac are the other than A intersections of the circle (P,PA) with the lines L1,L2

- the orthocenter of AAbAc

The perpendiculars to L1,L2 through P intersect L2,L1 at Ac,Ab, resp.

In general, P be a fixed point on the Euler line of AAbAc.
(ie PO/PH = m/n, where m,n given numbers)

Problem 2:

Let ABC be a triangle and AaBbCc the orthic triangle. Let M1 be a line intersecting AB,AC at Ab,Ac resp. such that Aa is a fixed X(i) point on the Euler line of AAbAc. Similarly M2 a line intersecting BC,BA at Bc,Ba resp. such that Bb is X(i) point on the Euler line of BbBcBa, and M3 a line intersecting CA,CB at Ca,Cb such that Cc is X(i) point on the Euler line of CcCaCb.

For which points X(i) the Euler Lines of the triangles:

1. AAbAc, BBcBa, CCaCb

2. AaAbAc, BbBcBa, CcCaCb

are concurrent?

[For X(i) = O is the Problem Concurrent Euler Lines]

APH, 22 February 2012