The square ABCD has sides of length 1 unit and it is split into three triangles by the lines BP and CP. If P is the midpoint of AD, find the radii of the inscribed circles of these triangles.

Now suppose the lengths AP and PD are (1- p) and p respectively. Find the radii of the three circles r 1,r 2 and r 3 in terms of p and plot, on the same axes, the graphs of r 1, r 2 and r 3 as p varies from 0 to 1. Can the ratio of the radii r 1 : r 2 : r 3 ever take the value 1:2:3 ?

square ABCD with three circles within it.

Sangaku in Japanese means a mathematics tablet. During the Edo period (1603-1867) when Japan was cut off from the western world, people of all classes produced theorems in Euclidean geometry as beautifully coloured drawings on wooden tablets to be hung in a temple. Proofs were rarely given. The tablets challenged other geometers: "See if you can prove this."