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Home » Circles » Tangent Lines » Tangent-Secant Theorem

Tangent-Secant Theorem

Last updated: Dec 14, 2021 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

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 AB2=AC · AD

Tangent-secant

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 lines ( AB2=AC · AD).

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

tangent-secant with triangles

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 AB2=AC · AD.

Proof

Here's the proof of the Tangent-Secant Theorem:

(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) AB2=AC · AD //(5), property of similar triangles

« The Tangent-Chord Theorem
Circumscribed Circle »

About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected]

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Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

Geometry Topics

  • Area of Geometric Shapes
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    • Triangle Inequalities

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