Dear Alexey

[AZ]

> >> I have a next hypothesis. Let A_1...A_n is the cyclic polygon.

Then

> >> 1) The Brocard points exist iff the symmedians of triangles

> >> A_{i-1}A_iA_{i+1} from A_i concur in point L.

[JPE]

> >And what do you think of the following conjecture :

> >A cyclic polygon has Brocard points if and only if the polygon is

> >the inverse of a regular polygon?

[AZ]

> I think that this two conditions are equivalent. If the point L

exists then

> the projective transformation T conserving the circumcircle and

such that

> T(L)=O transforms our polygon to regular. In the circle this

transformation

> coincide with some inversion.

Suppose that A_1,...,A_n has Brocard points P,Q; let M be a Poncelet

point of the pencil generated by the circumcircle (O) of the polygon

and the circle OPQ. the line MA_k intersects again (O) at B_k.

Then B_1,...,B_n is regular.

Of course, it is necessary to prove that such a point M is real, ie

that (O) and the circle OPQ cannot intersect.

It is necessary too to prove the reciprocal.

I think that it could be possible to name these two points M the

isodynamic points of the Brocardian polygon because

MA_(i+k)/M_A(i-k) = A_iA_(i+k)/A_iA_(i-k) for all i and k.

Friendly. Jean-Pierre