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Let ABC be a triangle with incircle $\gamma$ and circumcircle $\Gamma$. Let $\Omega$ be the circle tangent to rays $AB, AC,$ and to $\Gamma$ externally, and let $A^{\prime}$ be the tangency point of $\Omega$ with $\Gamma$. Let the tangents from $A^{\prime}$ to $\gamma$ intersect $\Gamma$ again at $B^{\prime}$ and $C^{\prime}$. Finally, let $X$ be the tangency point of the chord $B^{\prime}C^{\prime}$ with $\gamma$. Prove that the circumcircle of triangle $BXC$ is tangent to $\gamma$.

This is something I found from geogebra. Is there a proof of it? Thanks in advance.

shadow10
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  • What does mixtillinear mean? Top Google results point to this question. –  Jul 07 '14 at 00:35
  • $\Omega$ is known as the $A$-mixtillinear incircle. The circle touching the circumcircle internally, and two of the sides of the triangle is called the mixtillinear circle. Now depending on the vertices the name changes. The title may mislead you, but everything required to prove, is mentioned in the problem. I suppose I have cleared up your query a bit @Thisismuchhealthier. – shadow10 Jul 07 '14 at 12:18
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    The proper spelling is mixtilinear (one "l"), for which web-search results are plentiful. See, eg, Cut the Knot articles here and here, as well as MathWorld entries here, here, and here. An American Mathematical Monthly article is here. – Blue Jul 07 '14 at 13:32
  • Sorry for the mistake. I didn't know. Anyways, I hope it is more readable now. – shadow10 Jul 07 '14 at 14:23

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