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 5 of 5: Apply the tangent criterion twice
Detailed analysis
Since angle XYB=beta and angle BYZ=alpha, angle XYZ=alpha+beta. Thus angle BZX equals angle XYZ, so BZ (the same line as BW) is tangent to circle XYZ at Z. Also angle BXY=angle XYZ, so BX is tangent to the circle at X. Hence BW and BX are the required tangents.