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Home » Circles » Chords » Intersecting Secants Theorem

Intersecting Secants Theorem

Last updated: Mar 27, 2021 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

In today's lesson, we will present a detailed, step-by-step proof of the Intersecting Secants Theorem, using properties of similar triangles. This is a fairly simple proof, so today's lesson will be short.

A secant is a line that extends from a point outside the circle and goes through the circle. It intersects the circle at two points, and the line segment between those two points inside the circle is a chord.

Similar to the Intersecting Chords Theorem, the Intersecting Secants Theorem gives the relationship between the line segments formed by two intersecting secants.

Problem

AB and AC are two secant lines that intersect a circle. Show that AD⋅AB=AE⋅AC.

Geometry drawing: intersecting secants

Strategy

We are asked to show the relationship between line segment lengths as a product of their lengths. This is a clear hint to use triangle similarity, since we know that in similar triangles ΔABC∼ ΔDEF, AB*EF = BC*DE.

So let's draw a couple of triangles in which the sides AD, AB, AE and AC are sides:

 intersecting chords similarity

We can now easily show the two triangles ΔABE and ΔACD are similar (one shared angle, one pair of congruent angles which subtend the same arc), and the required relationship immediately follows.

Proof

(1) ∠BAC ≅ ∠CAB //Common angle to both triangles, reflexive property of equality
(2) ∠ABE ≅ ∠ACD // Inscribed angles which subtend the same arc are equal
(3) ∠BEA ≅ ∠CDA //(1), (2), Sum of angles in a triangle
(4) ΔABE ∼ ΔACD //angle-angle-angle
(5) AD⋅AB = AE⋅AC //(4), property of similar triangles

« Tangents to a circle and inscribed angles
The Tangent-Chord Theorem »

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
    • Chords
    • Inscribed Shapes
    • Tangent Lines
  • Lines and Angles
    • Intersecting Lines and Angles
    • Parallel Lines
    • Perpendicular lines
  • Pentagons and Hexagons
  • Perimeter of Geometric Shapes
  • Polygons
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    • Parallelograms
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  • Triangles
    • Congruent Triangles
    • Equilateral Triangles
    • Isosceles Triangles
    • Pythagorean Theorem
    • Right Triangles
    • Similar Triangles
    • Triangle Inequalities

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