# Thread: Stuck on proof of Secant-Tangent Theorem

1. ## Stuck on proof of Secant-Tangent Theorem

Secant-Tangent Theorem states: If a secant PA and tangent PC meet a circle at the respective points A, B, and C (point of contact),
then (PC)^2 = (PA)(PB).

My work so far on the proof:
Given circle O with secant PA and tangent PC which meet circle O at A, B, and C.
Draw chords AC and BC.
Angle CPA is congruent to Angle CPB.
Angle PCA is congruent to Angle PBC (both angles intercept arc AC).
Triangle ACP ~ Triangle DBP by the Angle-Angle Similarity Criterion.

Now, this is where I am stuck. I know the two triangles are similar. My hunch is to manipulate the relations of corresponding parts of similar triangles to get to what I want to prove that
(PC)^2 = (PA)(PB).

Will the Inscribed Angle Theorem help any?
(Inscribed Angle Theorem: The measure of an inscribed angel of a circle equals 1/2 that of its intercepted arc.)

Any help is greatly appreciated! Thank you for your time.

2. $\Delta ACP\sim\Delta CBP\Rightarrow \frac{PC}{PB}=\frac{PA}{PC}\Rightarrow PC^2=PA\cdot PB$

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# prove secant tangent theorem

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