Question

Discover and prove a theorem about two lines tangent to a circle at the endpoints of a diameter.

Short answer

Lines are parallel.

Step-by-step solution

Step 1. Given information.

Let a circle with centre $O$ and diameter $CD$. Let $MN$ be the tangent at point $C$ and $PQ$ be the tangent at point $D$.

Step 2. Formula used.

Tangent at any point of circle is perpendicular to the radius through point of contact.

Step 3. Proof.

Consider the figure below,

Here, $MN$is tangent at point C and $PQ$is tangent at point D. Therefore,

$\begin{array}{l}OC\perp MN\\ OD\perp PQ\end{array}$

Therefore,

$\begin{array}{l}\angle OCN=90°\\ \angle ODQ=90°\end{array}$

i.e., $\angle OCM=90{}^{\circ }\text{and}\angle ODP=90°$

For lines $MN$and $PQ$and transversal $AB$, $\angle DCM=\angle CDQ$,

both alternate angels are equal.

Therefore, lines are parallel.