Food for thought

## Solution

In this interactivity, you can move the points $A$, $B$, $C$ and $D$ around the circumference of the circle, with $D$ between $B$ and $C$.

1. Three angles are marked.

What can you say about the angles at $C$ and $D$?

What happens to these angles if you move $C$ or $D$, keeping $A$ and $B$ fixed?

How do you know this?

In this situation, we have a fixed chord $AB$ and the angles at the circumference $\angle ACB$ and $\angle ADB$ on the same side of the chord. This reminds us of the theorem that angles at the circumference on the same side of the same chord are equal. Therefore $\angle ACB$ is constant regardless of where $C$ lies, and likewise $\angle ADB$ is constant. Furthermore, because of this theorem, they are also equal: $\angle ACB=\angle ADB$.

1. What happens to the exterior angle at $B$ (shown in blue) as $D$ approaches $B$?

We have labelled the angle at $B$ as $\theta$. This makes it easier to talk about it. It is equal to the sum of the opposite two interior angles of the triangle $ABD$, that is $\theta=\angle BAD+\angle ADB$.

Why is this the case? It is because of two standard angle theorems:

1. $\angle BAD+\angle ADB+\angle ABD=180^\circ$, since the angles in a triangle sum to $180^\circ$.

2. $\theta+\angle ABD=180^\circ$, because angles on a straight line sum to $180^\circ$.

Then equating these expressions and subtracting $\angle ABD$ shows that $\theta=\angle BAD+\angle ADB$.

As the point $D$ moves towards $B$, the angle $\angle BAD$ tends towards zero, and so $\theta$ tends to $\angle ADB$, which is the same as $\angle ACB$.

Alternatively, we might say that as $D$ moves towards $B$, the chord $AD$ tends to the chord $AB$, so the angles $\angle ADB$ and $\theta$ become closer and closer to each other. Since $\angle ADB=\angle ACB$ remains fixed, the exterior angle $\theta$ tends towards $\angle ACB$.

1. Which other circle theorem can you deduce from this?

Exploring the interactivity seems to suggest that the line $BD$ becomes closer and closer to the tangent to the circle at $B$. This is, indeed, the case; we explore tangents in more detail at Calculus of Powers.

We can say that in the limit, as $D$ approaches $B$, the line $BD$ tends to the tangent to the circle at $B$.

Also, as we have just observed, in the limit, the exterior angle $\theta$ tends to $\angle ACB$. On the other hand, the line $BD$ tends to the tangent at $B$, so the angle $\theta$ tends to the angle between the tangent at $B$ and the chord $BA$.

So in the limit, we see that the (exterior) angle between $BA$ and the tangent at $B$ equals $\angle ACB$.

This is known as the alternate segment theorem.