Using the Tangent-Chord Theorem, it is simple to prove the third theorem which provides a relationship between lines in circles – the Tangent-Secant Theorem (the other two being the Intersecting Secants Theorem and the Intersecting Chords Theorem).

## Problem

AB is tangent to circle O, and AC is a secant line intersecting the circle at points C and D. Prove that AB^{2}=AC · AD

## Strategy

Just as we did in the other two theorems which provide a relationship between the lengths of line segments of tangents or chords, we are pointed in the direction of using triangle similarity by the fact that we need to prove a relationship involving the product of lengths of liens ( AB^{2}=AC · AD).

So let’s construct triangles in which AB, AC and AD are sides, and show they are similar:

Looking at triangles ΔABC and ΔADB, we see they share one angle, ∠BAC. Using the Tangent-Chord Theorem, we see that ∠ABC ≅ ∠ADB, so the two triangles are similar, and it follows that AB^{2}=AC · AD.

## Proof

(1) ∠BAC ≅ ∠BAC //Common angle

(2) ∠ABC ≅ ∠ADB //Tangent-Chord Theorem

(3) ∠ACB ≅ ∠ABD //Sum of Angles in a Triangle

(4) ΔABC∼ ΔADB //Angle-Angle-Angle

(5) AB^{2}=AC · AD //(5), property of similar triangles