Two circles touch the line AB at A and B and intersect each other at X and Y with X nearer to the line AB. The tangent to the circle AXY at X meets the circle BXY at W. The ray AX meets BW at Z. Show that BW and BX are tangents to the circle XYZ.
Solution
Let angle ZXW = a and angle ZWX = b. XW is tangent to circle AXY at X, so angle AYX = a. AB is tangent to circle AXY at A, so angle BAX = a. AB is tangent to circle BXY at B, so angle ABX = b. Thus, considering triangle ABX, angle BXZ = a+b. Considering triangle ZXW, angle BZX = a+b.
BXYW is cyclic, so angle BYX = angle BWX = b. Hence angle AYB = angle AYX + angle XYB = a+b = angle AZB. So AYZB is cyclic. Hence angle BYZ = angle BAZ = a. So angle XYZ = angle XYB + angle BYZ = a+b. Hence angle BZX = angle XYZ, so BZ is tangent to circle XYZ at Z. Similarly angle BXY = angle XYZ, so BX is tangent to circle XYZ at X.
(C) John Scholes
jscholes@kalva.demon.co.uk
11 Apr 2002