### In the given figure, two circles touch each other at a point $C$. Prove that the common tangent to the circles at $C$ bisects the common tangent at the points $P$ and $Q$. C A B P Q R

Step by Step Explanation:
1. We see that $PR$ and $CR$ are the tangents drawn from an external point $R$ on the circle with center $A$.
Thus, $PR = CR \space \space \space \ldots \text{(i)}$

Also, $QR$ and $CR$ are the tangents drawn from an external point $R$ on the circle with center $B$.
Thus, $QR = CR \space \space \space \ldots \text{(ii)}$
2. From $eq \space \text{(i)}$ and $eq \space \text{(ii)}$, we get
$PR = QR \space \space \text{ [Both are equal to CR]}$

Therefore, $R$ is the midpoint of $PQ$.
3. Thus, we can say that the common tangent to the circles at $C$ bisects the common tangent at the points $P$ and $Q$.