Start with a circle and draw two tangent circles inside. The (black) inner tangent lines to the smaller circles intersect the large circle. The (red) lines through these intersection points are parallel to the (green) outer tangents to the small circles.

A long time ago I worked on this theorem, but I never knew the name. Without a name it's difficult to find more information. Does anyone know if this theorem has a name and where I can find more information about it?

Image of the theorem

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    $\begingroup$ If it doesn't have a name, you can rightfully call it "angry dude" (or some variation thereof) :-) $\endgroup$ – M.G. Oct 26 '17 at 16:48
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    $\begingroup$ This reminds me of the following problem from an old issue of "Kvant" (around mid-80s). Consider an inscribed quadrilateral ABCD where neither pair of the opposite sides are parallel. Now form the triangle T whose vertices are intersections of lines AB and CD, AC and BD, AD and BC. Then the center of the circle in which ABCD is inscribed is the orthocenter of T. I remember that the proof had to do with poles and polars. It's probably all online, not too hard to find. $\endgroup$ – Alexander Burstein Oct 27 '17 at 1:33
  • $\begingroup$ Hyperbolic Hyperbole Theorem in Two-Dimensions? $\endgroup$ – user102126 Jul 5 '18 at 7:40
  • $\begingroup$ Please see similarly subject: mathoverflow.net/questions/304300/… $\endgroup$ – Đào Thanh Oai Apr 24 at 11:18

Even more is true for this theorem. Check out this drawing from Arseniy Akopyan wonderful book of Geometry in Figures (Second, extended edition, 2017). On page 65 we find Figure 4.7.29)

Figure 4.7.29)

In the foreword, Arseniy Akopyan writes

"It is commonly very hard to determine who the author of a certain result is."

He nevertheless provides source for many of the figures in the end of his book. Unfortunately for Figure 4.7.29 he doesn't provide such a reference.

This leads me to the answer: Probably it doesn't have a name (like many "geometry theorems").

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    $\begingroup$ Is the inner circle in this diagram any circle tangent to the two smaller circles, or is it a specific one? $\endgroup$ – Carl Schildkraut Oct 26 '17 at 20:13
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    $\begingroup$ It's any circle tangent to both, just as you can replace the outer circle by any circle tangent to both. $\endgroup$ – David Eppstein Oct 27 '17 at 5:35
  • $\begingroup$ If you draw a circle that is tangent to both of the fixed circles, but circumscribes only one of them, then look at the points where it intersects the two green tangents from the original post (dashed in this picture), would that make two chords that are parallel to the black / solid tangent lines? (I.e. swap the roles of the two pairs of tangent lines.) $\endgroup$ – Arthur Oct 27 '17 at 8:05
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    $\begingroup$ @Arthur thanks, I was too excited by the diagrams to read the comments! $\endgroup$ – user1998586 Oct 27 '17 at 14:30
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    $\begingroup$ I think, I've found the statement by myself, looking for a static version of Protasov's segment theorem demonstrations.wolfram.com/SegmentTheorem $\endgroup$ – Arseniy Akopyan Nov 6 '17 at 11:43

This is theorem 2 (the Parallel tangent theorem) in "Two Applications of the Generalized Ptolemy Theorem" by Shay Gueron.


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