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If a tangent and a secant intersect in the exterior of a circle, then the square of the measure of the tangent is equal to the product of the measures of the secant and its external secant segment.
In the above diagram, JK is tangent and JM is secant to the circle. Then the following statement holds true.
JK^2 = JL * JM
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.
JK^2 = JL * JM