Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.7 AIME, harder Prove it

Let $\Gamma$ be a circle, $[A B]$ a chord of this circle, and $\omega_{1}, \omega_{2}$ two circles internally tangent to $\Gamma$ at $T_{1}$ and $T_{2}$ , and to $(A B)$ at $T_{1}^{\prime}$ and $T_{2}^{\prime}$ , such that $T_{1}$ and $T_{2}$ are on the same side of $[A B]$ . These two circles intersect at $C$ and $D$ . Show that $(C D)$ passes through the midpoint of the arc $\widehat{A B}$ not containing $T_{1}$ .

Solution

Using the previous exercise, and introducing $S$ as the midpoint of the arc $\widehat{A B}$ not containing $T_{1}$ , we have that $T_{1}, T_{1}^{\prime}, S$ are collinear, and $T_{2}, T_{2}^{\prime}, S$ are also collinear. Then, according to the property of the South Pole, $T_{1}, T_{2}, T_{1}^{\prime}, T_{2}^{\prime}$ are concyclic. We can convince ourselves of this by calculating the value of the intercepted arcs:

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.