Dear Alexey,

I wrote

> suppose that ABC is a variable isosceles triangle (in B) inscribed

> in a fixed circle with center o; let M be a fixed point and A'B'C'

> the cevian triangle of M wrt ABC.

Of course, I was meaning the circumcevian triangle of M wrt ABC.

Sorry for the typo.

> It is very easy to check that the

> B'-symedian of A'B'C' intersects the line OM at a point L

> independant of the choice of ABC.

> If S is a point of the circle such as OM and OS are perpendicular,

L

> is the projection upon OM of the reflection of O wrt SM. More over

L

> lies on the half-line OM and inside the circle.

> From this it follows that if B_1,...,B_n is a regular polygon

> inscribed in the circle, if A_k is the second intersection of MB_k

> with the circle, every A_k symedian of A_(k-1),A_k,A_(k+) goes

> through L.

> That's probably a part of the proof you're waiting for.

> Friendly. Jean-Pierre