Complex Tangent Systems

The Power of Combining Theorems

Many complex circle problems can't be solved with a single rule. They require combining Theorem 10.1 (tangent is perpendicular to radius) and Theorem 10.2 (tangents from an external point are equal).

Problem. Prove that in two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.

Given: Two concentric circles $C_1$ and $C_2$ with center $O$, and a chord $AB$ of the larger circle $C_1$ which touches the smaller circle $C_2$ at point $P$.

Problem. Two tangents $TP$ and $TQ$ are drawn to a circle with center $O$ from an external point $T$. Prove that $\angle PTQ = 2 \angle OPQ$.

Given: A circle with center $O$, an external point $T$, and two tangents $TP$ and $TQ$ to the circle, where $P$ and $Q$ are the points of contact.

Problem. $PQ$ is a chord of length $8\text{ cm}$ of a circle of radius $5\text{ cm}$. The tangents at $P$ and $Q$ intersect at a point $T$. Find the length $TP$.