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Home » Circles » Tangent Lines » The Two Tangent Theorem

The Two Tangent Theorem

Last updated: May 1, 2024 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

This theorem states that if we draw two lines from the same point which lies outside a circle, such that both lines are tangent to the circle, then their lengths are the same. We will now prove that theorem.

Problem

AB and AC are tangent to circle O. Show that AB=AC

two tangent theorem in geometry

Strategy

To show two lines are equal, a helpful tool is triangle congruency. The shape of the drawing in the problem statement - where the two tangent lines create a sort of triangular 'clown's hat' also suggests we would be served by constructing some triangles here, where the two tangent lines are sides.

We also know that a property of the tangent lines to a circle is that they form a 90° angle between the line and a radius at the point of tangency - so let's draw that:

Geometry: two radii

And now the triangles almost present themselves. Connect the point A with a line segment to the center of the circle, O, and we will have two right triangles, with a common hypotenuse (AO), and an equal leg, as both radii are equal, and the triangles are congruent by HL.

Geometry shape: tangents with triangles

Proof

(1) AB is tangent to Circle O     //Given
(2) ∠ABO=90°                          //tangent line is perpendicular to circle
(3) AC is tangent to Circle O     //Given
(4) ∠ACO=90°                          //tangent line  is perpendicular to circle
(5) AO=AO                              //common side (reflexive property)
(6) OC=OB=r                           //radii of a circle are all equal
(7) △ABO≅△ACO                 //Hypotenuse-leg 
(8) AB=AC                             // Corresponding sides in congruent triangles (CPCTC) 

This theorem is a key element in proving the Pitot Theorem, about a quadrilateral that circumscribes a circle.

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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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About

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
  • Circles
    • Arcs, Angles, and Sectors
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    • Intersecting Lines and Angles
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  • Triangles
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    • Equilateral Triangles
    • Isosceles Triangles
    • Pythagorean Theorem
    • Right Triangles
    • Similar Triangles
    • Triangle Inequalities

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