How to Prove the Intersecting Chords Theorem of Euclid

Step-by-Step Guide

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   Step 1: Understand a definition of Euclid's Intersecting Chords Theorem.

   The Intersecting Chords Theorem asserts the following very useful fact:
   Given a point P in the interior of a circle with two lines passing through P, AD and BC, then AP*PD = BP*PC
   -- the two rectangles formed by the adjoining segments are, in fact, equal.
   
   This article shows you in a few steps how to prove this is true.
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   Step 2: Prove the similarity of triangles ABP and CDP that is a consequence of their angles since: BAD = BCD because inscribed angles subtended by the same chord BD are equal ; ABC = ADC because inscribed angles subtended by the same chord AC are equal ; and APB = CPD because they are a pair of vertical angles (vertical angles are formed by the same intersecting lines).

   , That is fundamentally how similar triangles are related. , That is how the Theorem was arrived at, both geometrically and mathematically, for these two products are indeed rectangles. , To understand how these proofs operate, you are referred to the translated text of Euclid's "Elements" below. ,, For more art charts and graphs, you might also want to click on Category:
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   Step 3: Prove that from the similarity of triangles ABP and CDP are obtained these identities and proportions: 1) AP/PC = BP/PD = AB/CD.

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   Step 4: Prove that the first identity above

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   Step 5: AP/PC = BP/PD

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   Step 6: leads directly to the Intersecting Chords Theorem

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   Step 7: by cross-multiplying: AP*PD = BP*PC.

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   Step 8: Research and find out that the proof given by Euclid is much longer and more involved

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   Step 9: and uses the Pythagorean Theorem

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    Step 10: which is a fairly lengthy proof in itself.

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    Step 11: Make use of helper articles when proceeding through this tutorial: See the article How To Multiply and Divide Geometrically Like Mother Nature for a list of articles related to Excel

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    Step 12: Geometric and/or Trigonometric Art

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    Step 13: Charting/Diagramming and Algebraic Formulation.

Detailed Guide

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