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Home » Circles » Chords » Intersecting Chords Theorem

Intersecting Chords Theorem

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

The intersecting chords theorem states that when two chords intersect at a point, P, the product of their respective partial segments is equal.

In other words: AP*PB=CP*PD

Problem

Prove that when two chords intersect in a circle, the products of the lengths of the line segments on each chord are equal.

Intersecting chords in geometry

Strategy

There are two hints given in the problem statement. The first hint is that it asks to show an equality of the product of two line segments.

This should immediately bring to mind a relationship we learned when studying similar triangles - which is that in similar triangles there is a relationship between the product of line segments- if ΔABC∼ ΔDEF, then AB*EF = BC*DE

So we will need to draw triangles where these line segments (AP, PB, CP, PD) are sides of the triangles, and then show the triangles are similar.

Let's draw it:

Geometry: intersecting chords with similar triangles

But how will we prove the triangles are similar? The second hint is that we are dealing with chords. And when we drew the lines that form the triangles (AC and DB) we created several inscribed angles that are on the same arcs. So by the inscribed angle theorem, they are equal.

We will now show that the triangles are similar by showing all their angles are equal, and we are done.

Proof

Here's how you prove the Intersecting Chords Theorem:

(1) ∠CAB≅∠BDC      //inscribed angle theorem, both subtend the arc CB
(2) ∠ACD≅∠ABD      //inscribed angle theorem, both subtend the arc AD
(3) ∠APC≅∠DPB      //Vertical angles, or alternatively, third pair of angles in a triangle where the other two pairs are equal
(4) ΔACP ∼ ΔDBP   //Angle-Angle-Angle
(5) AP*PB = DP*PC //corresponding line segments in similar triangles

« Proving the Inscribed Angle Theorem
Polygon Inscribed in a 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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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
  • Quadrangles
    • Kites (Deltoids)
    • Parallelograms
    • Rectangles
    • Rhombus
    • Squares
    • Trapezoids
  • Triangles
    • Congruent Triangles
    • Equilateral Triangles
    • Isosceles Triangles
    • Pythagorean Theorem
    • Right Triangles
    • Similar Triangles
    • Triangle Inequalities

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