I was already studying about Poncelet porism but unfortunately I couldn't find any useful thing about this theorem for two intersecting circles. even I don't know if it is true for intersecting circles .

I draw some pictures using GeoGebra and I find out after some finite steps of drawing tangents ($n$) the last tangency point will be one of the meeting points of circles therefore ($n+1$)th and $n$th tangent would be coincident and then ($n+2$)th and ($n-2)$th tangent would be coincident and the process follows till the ($2n$)th tangent coincide the first one so it seems the theorem is also true for intersecting circles but I can't find any mathematical proof for it. I tried to prove it using inversion like when two circles lie entirely one within the other but it is not possible to find an inversion such that circles become concentric.

please let me know if it this theorem is also true for meeting circles and how can I prove it or recommend some sources or links for this beautiful theorem. thanks!

edit: I just download an essay here according to link if there are two circles satisfying the formula $d^2=R^2-2Rr$ where $R,r$ are radius of circles and $d$ is distance between their centers, then there are infinitely many triangles inscribed in one and circumscribed about the other circle. how can I prove it?

this link may also help.

please let us to consider and solve the last part first. thanks in advance

  • $\begingroup$ I would like to see what happens if one goes through the proof in my bachelor's thesis: An explicit algebro-geometric proof of Poncelet's closure theorem. What are the equations of your circles? $\endgroup$ – Ricardo Buring Jan 16 '19 at 21:33
  • $\begingroup$ Take the simplest case: a triangle. Specifically, consider $\triangle ABC$, its circumcircle, and one of its excircles (say, the one opposite $A$). Let $A^\prime$ be a point on the circumcircle (but outside the excircle), and let the tangents through $A$ to the excircle meet the circumcircle at $B^\prime$ and $C^\prime$. A GeoGebra sketch suggests (though I have not yet formally proven) that $\overline{B^\prime C^\prime}$ is also tangent to the excircle, which (if true) validates the "intersecting Poncelet" notion for triangles. $\endgroup$ – Blue Jan 17 '19 at 6:01
  • $\begingroup$ @Blue see here for this case : cut-the-knot.org/Curriculum/Geometry/Poncelet3E.shtml $\endgroup$ – math enthusiastic Jan 17 '19 at 6:08
  • $\begingroup$ @RicardoBuring you mean this theorem can be true for meeting circles with specific equation not all of them? I just try it for two arbitrary circles $\endgroup$ – math enthusiastic Jan 17 '19 at 6:08
  • $\begingroup$ @mathenthusiastic: I had a feeling the excircle result was known. :) As to your question to Ricardo: Note that Poncelet says "If a path closes for one starting point, then it closes for any starting point." Poncelet does not guarantee a closing path for arbitrary circles. $\endgroup$ – Blue Jan 17 '19 at 6:13

According to the final edit in the question the more pressing request seems to be for a proof of how there are an infinite number of bicentric triangles for two nested circles satisfying the formula $d^2=R^2-2Rr$.

A proof is given at Cut the Knot's Euler's Formula and Poncelet Porism. Conveniently, the case for intersecting circles satisfying $d^2=R^2+2Rr$ is also covered.

For bicentric polygons, Poncelet's theorem covers the case of two conics in general position. So that would cover the case of intersecting conics and in particular intersecting circles. I'd recommend taking a look at Halbeisen and Hungerbuhler's A Simple Proof of Poncelet’s Theorem (on the occasion of its bicentennial) for some background and proofs. There is also a sketch of a proof using elliptic curves on Wikipedia.


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