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 4 of 5: Create a second cyclic quadrilateral
Detailed analysis
The points B,X,Y,W are cyclic, so angle BYX equals angle BWX=beta. Therefore angle AYB=alpha+beta=angle AZB, and A,Y,Z,B are cyclic. It follows that angle BYZ equals angle BAZ=alpha.