Problem of the Week 1121

A Geometrical Alignment

Consider two semicircles S and T that emanate from the same point with diameters along a common line, with S being the larger.

Draw an isosceles triangle whose base is the part of diameter of S that is outside T and whose apex is on S. Draw a circle inside S that is tangent to S, T, and the triangle.

Prove that the center of this circle is directly above the point common to the triangle and T.

Source: Suggested by Sandy Kline, Seattle, Wash. This is a Japanese Temple Problem from 1803. Reference: "Japanese Temple Geometry Problems," San Gaku, H. Fukagawa and D. Pedoe, Charles Babbage Research Centre, Winnipeg, 1989.

© Copyright 2009 Stan Wagon. Reproduced with permission.