China Western Mathematical Olympiad

Question

The circles O_1 and O_2 meet at points A and B. The line DC passes through O_1, intersects the circle O_1 at D and is a tangent to the circle O_2 at C. Also, CA is a tangent to the circle O_1 at A. The secant AE of the circle O_1 is perpendicular to DC. AF is perpendicular to DE and meets DE at F. Prove that BD bisects the line segment AF. (posed by Bian Hongping) FIGURE_0

Accepted Answer

Let AE intersect DC at point H, FIGURE_0 and AF intersect BD at point G. Join AB, BC, BH, BE, CE and GH. By symmetry, CE is also a tangent line of the circle O_1 and H is the midpoint of AE. Since AF DE, we have AGB = 2 - BDE. By AE DC, AHB = 2 - BHC. By ①, ② and ③, AGB = AHB. Therefore A, G, H, B lie on the same circle, and AHG = ABG = AED. Thus GH DE. Since H is the midpoint of AE, G is the midpoint of AF.