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Segments of Secants and Tangents Theorem

Rule

Segments of Secants and Tangents Theorem

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, is tangent to the circle and is secant.

Proof

Consider the auxiliary segments and

By the Inscribed Angle Theorem, the measure of is half the measure of its intercepted arc which is Since is a tangent and is a chord, the Tangent and Intersected Chord Theorem can be applied to obtain the following equation. The last two equations imply that and then In addition, the Reflexive Property of Congruence gives that
Separate Triangles

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