### Circles Ad Infinitum

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

### So Big

One side of a triangle is divided into segments of length a and b by the inscribed circle, with radius r. Prove that the area is: abr(a+b)/ab-r^2

### Incircles Explained

This article is about triangles in which the lengths of the sides and the radii of the inscribed circles are all whole numbers.

# Sangaku

##### Stage: 5 Challenge Level:

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 $1:2:3$?

Notes and Background

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."