Dear Ivan and François!
I found the proof. Consider projective map transforming the inconic to the circle and the concurrence point to its center. It is easy to see that the image of given polygon will be regular. Thus the circumconic will be the concentric circle.
I do not consider triangles, I consider 2k+1 - gons with k > 1,
so the smallest case is pentagon. In the case of a triangle,
indeed the locus of Gergonne point is a circle when triangle
moves in the porism. If you consider the pentagon case, the problem is more interesting.
A nice problem is to draw a pentagon that is circumscribed about a circle such that its gergonians are concurrent. Then there is a conic passing through five vertices of this pentagon.
Now, for any other pentagon in the porism of this conic and initial circle the gergonian must be concurrent at the same point.
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