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# BMO Geometry Question Watch

1. Two circles are internally tangent at a point T. A chord AB of the outer circle touches the inner circle at the point P. Prove that TP bisects angle ATB.

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2. Call the point where BT meets the smaller circle N and the point where AT meets the smaller circle M. Note that MTNP is cyclic. The result follows after a bit of angle chasing using alternate segment theorem and the cyclic quadrilateral property a few times.
3. (Original post by Unbounded)
Call the point where BT meets the smaller circle N and the point where AT meets the smaller circle M. Note that MTNP is cyclic. The result follows after a bit of angle chasing using alternate segment theorem and the cyclic quadrilateral property a few times.
But you assume that BA is tangential to the smaller circle whereas it is not given in the question.
4. (Original post by The Wayfarer)
But you assume that BA is tangential to the smaller circle whereas it is not given in the question.
"Touches" means it only passes through the circle once, i.e. is tangent to the circle.

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