## Problem

We take a circle and inscribe a triangle in it.

Complete the steps of the table below to find an interesting relationship between the angles and sides of this triangle.

Diagram Description
1. Add a point $C'$ on the circumference of the circle, so that $\angle C'AB$ is a right-angle.

2. What is the relationship between angles $ACB$ and $AC'B$

3. What is the length $C'B$?
4. Label side $AB$ as $c$. Write a relationship between $c$ and the angle at $C'$.
5. Can you therefore write a relationship between $c$ and the angle at $C$?
6. Starting from the original diagram, add a point $B'$ on the circumference of the circle, so that $\angle B'CA$ is a right-angle.

7. What is the relationship between angles $ABC$ and $AB'C$?

8. What is the length $B'A$?
9. Label side $AC$ as $b$. Write a relationship between $b$ and the angle at $B'$.
10. Therefore write a relationship between $b$ and the angle at $B$.
11. In general, what can we now say about the relationship between the sides and angles of a triangle?

You have proved your statement for this particular triangle, which has its three vertices on the circumference of a circle. Is your statement true for all triangles?

If you are not sure, you might like to look at Finding circles.