The porismcircles

The midcircle of two circles is the circle that swaps these two circles by inversion. For some background (in Ducth) see middencirkel by Dick Klingens. This inversion also maps the sides of ABC to circles. These circles we call the porismcircles.

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Poncelet's porism

The term "Poncelet's porism" points to the (limited) closure theorem by Poncelet. In short it says that to a given triangle with incircle and circumcircle, there are infinitely many triangles with the same incircle and circumcirlce. Each point of the circumcircle is vertex of such a triangle. See for instance (in Dutch) Om- en incirkel by Dick Klingens.

Inversion in the midcircle of incircle and circumcircle maps the sides of a triangle to a triple of circles tangent to the circumcircle and intersecting on the incircle. I call them porismcircles. They are the circular equivalent of the poristic triangles.

If we take another triangel with the same incircle and circumcircle, the X55 and X57 stay unchanged.

Thanks to Peter Moses for his assistance on these results.

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