Segments of Secants and Tangents Theorem

Consider the auxiliary segments KL and KM.

By the Inscribed Angle Theorem, the measure of ∠ M is half the measure of its intercepted arc KL. m∠ M = 1/2mKL Note that JK is a tangent to the circle and KL a chord. Therefore, by the Tangent and Intersected Chord Theorem, it can be said that the measure of ∠ JKL is half the measure of KL. m∠JKL = 1/2mKL By the Transitive Property of Equality, it can be stated that ∠ M and ∠ JKL have the same measure. Therefore, they are congruent angles. Additionally, by the Reflexive Property of Congruence, ∠ J is congruent to itself.

m∠ M = 1/2mKL [0.2cm] m∠JKL = 1/2mKL ⇓ m∠ M=m∠ JKL ⇓ ∠ M≅ ∠ JKL

∠ J≅ ∠ 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. JK/JM = JL/JK Lastly, by cross multiplying, the desired result is obtained.