Below are given two circles (with centers O and P) tangent to each other at A, and a line tangent to both circles at B and C, respectively.
Prove: that m<BAC = 90, in two completely different ways.
Circle Proofs

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    Method 1.
    The segment OA is perpendicular to the tangent at A.
    So is the segment PA. Therefore the segments OA and AP form a line OP.
    Consider the triangle OAB. Call the angle at A alpha. Since OAB is isosceles, alpha
    is also the angle at B. Since BC is tangent at B, the angle at B: ABC = 90-alpha.
    Consider the triangle PAC. It is also isosceles. Call the angle at A beta....
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