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.
- The center of the midcircle is the internal similitude center of the incircle and circumcircle,
in Kimberling's ETC
X55.
This point lies on the line through O (circumcenter) and I (incenter). It is also the common point
of the porismcircles and the point of intersection of the lines connecting the vertices of
ABC and the intersection points on the incircle.
- The lines connecting the vertic of ABC with the points of tangency of the porismcircles and the circumcirle
intersect in a point on OI, in Kimberling's ETC X57.
- When R and r are the circumradius and inradius and
RA, RB and RC the radii of the porismcircles, then we have the following equation:
.
- In particular we have
.
- Triangle KAKBKC is perspective to ABC. Thee perspector is
X955 from Kimberling's ETC.
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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