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Problem 3

Two circles touch the line AB at A and B and intersect at X and Y, with X nearer to AB. The tangent to the circle AXY at X meets the circle BXY at W. The ray AX meets BW at Z. Prove that BW and BX are tangents to the circle XYZ.
Step 2 of 5: Transfer tangent-chord angles
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∠AYX=∠BAX=α,∠ABX=β\angle AYX=\angle BAX=\alpha,\qquad \angle ABX=\beta
Detailed analysis

Because XW is tangent to circle AXY, angle AYX equals alpha. Since AB is tangent at A, angle BAX also equals alpha. Since AB is tangent at B and BW is the line through Z and W, angle ABX equals beta.