NICE GEOMETRY.
Let triangle ABC have a circumcircle L, and let M be the circle that touches the sides AB and AC and also touches L internally. Let the points of contact of M with AB and AC be P and Q, respectively, and let I be the incentre of triangle ABC. The point I divides PQ in what ratio. That is find the ratio \frac{PI}{IQ}.
The answer is 1.
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