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{{ printedBook.courseTrack.name }} {{ printedBook.name }} 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 diagram above, $JK$ is tangent to the circle and $JM$ is secant.

$JK_{2}=JL⋅JM$

Consider the auxiliary segments $KL$ and $KM.$

By the Inscribed Angle Theorem, the measure of $∠M$ is half the measure of its intercepted arc which is $KL.$ $m∠M=21 mKL $ Since $JK$ is a tangent and $KL$ is a chord, the Tangent and Intersected Chord Theorem can be applied to obtain the following equation. $m∠JKL=21 mKL $ The last two equations imply that $m∠M=m∠JKL$ and then $∠M≅∠JKL.$ In addition, theSeparate Triangles

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Notice that two angles of $△JKL$ are congruent to two angles of $△JMK.$ Therefore, the Angle-Angle Similarity Theorem gives that $△JKL∼△JMK.$ In consequence, the following proportion can be set. $JMJK =JKJL $ Finally, by cross multiplying, the desired result is obtained.

$JK_{2}=JL⋅JM$