Segments of Secants and Tangents Theorem

Consider the auxiliary segments $\overline{KL}$ and $\overline{KM}.$

By the Inscribed Angle Theorem, the measure of $\angle M$ is half the measure of its intercepted arc $\overgroup{KL}.$ Note that $\overline{JK}$ is a tangent to the circle and $\overline{KL}$ a chord. Therefore, by the Tangent and Intersected Chord Theorem, it can be said that the measure of $\angle JKL$ is half the measure of $\overgroup{KL}.$ By the Transitive Property of Equality, it can be stated that $\angle M$ and $\angle JKL$ have the same measure. Therefore, they are congruent angles. Additionally, by the Reflexive Property of Congruence, $\angle J$ is congruent to itself.

$$\begin{array}{c}\{\begin{array}{l}m\angle M=\frac{1}{2}m\overgroup{KL}\\ m\angle JKL=\frac{1}{2}m\overgroup{KL}\end{array}\\ \Downarrow \\ m\angle M=m\angle JKL\\ \Downarrow \\ \angle M\cong \angle JKL\end{array}$$

$\angle J\cong \angle J$

The above information can be visualized in the diagram. Notice that two angles of $△JKL$ are congruent to two angles of $△JMK.$ Therefore, by the Angle-Angle Similarity Theorem it can be stated that $△JKL$ and $△JMK$ are similar triangles. Consequently, the following proportion can be set. Lastly, by cross multiplying, the desired result is obtained.