# Sangaku

Problem:

We denote by h the altitude from the base of the right triangle in the picture, R the radius of the black half-circle and by r the radius of red half-circle. Express the ratio r/R in terms of R and h.

Hint : The radius R of an inscribed circle in a triangle is given by R = 2S/p where S is the area of the triangle and p its perimeter.

*From Sangaku. Le mystère des énigmes géométriques japonaises, Géry Huvent, Dunod, 2008*

The ratio is : r / R = R^{2} / h^{2}