Dear Darij!

>

>The equivalence of the conditions 2. and 5. was the

>subject of Problem 4 in the Bundeswettbewerb

>Mathematik (German National Mathematics Olympiad)

>2002, 2 round. In fact, it was a very hard problem;

>I was lucky enough to know the method of trilinear

>coordinates to solve it!

>

The equivalence of 2 and 5 is a particular case of next fact. Let given a

triangle ABC and a point P. A1, B1 are the common point of AP and BC, BP and

AC. Q - a point in A1B1, Q' is isogonally conjugated to Q, A2, B2 are the

common points of AQ' and BC, BQ' and AC. Then the point P' isogonally

conjugated to P is in A2B2. This can be easily proved by trilinear

coordinates.

Sincerely Alexey