Prove the Intersecting Chords Theorem of Euclid

When one first reads the Proposition 35 of Book III of Euclid's "Elements", one may be astounded that crossing chords create two equal rectangles, whether their intersection point is in the center or not, but it is fairly easy to understand. This article will teach you to prove the Intersecting (or Crossing) Chords Theorem; specifically, how the two chords AD and BC create two equal rectangles.

Steps

The Tutorial

  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.
  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 [Book III Propositions 20 and 21];
    • ABC = ADC because inscribed angles subtended by the same chord AC are equal [Book III Propositions 20 and 21]; and
    • APB = CPD because they are a pair of vertical angles (vertical angles are formed by the same intersecting lines).
  3. Prove that from the similarity of triangles ABP and CDP are obtained these identities and proportions: 1) AP/PC = BP/PD = AB/CD. That is fundamentally how similar triangles are related.
  4. Prove that the first identity above, AP/PC = BP/PD, leads directly to the Intersecting Chords Theorem, by cross-multiplying: AP*PD = BP*PC. That is how the Theorem was arrived at, both geometrically and mathematically, for these two products are indeed rectangles.
  5. Research and find out that the proof given by Euclid is much longer and more involved, and uses the Pythagorean Theorem, which is a fairly lengthy proof in itself. To understand how these proofs operate, you are referred to the translated text of Euclid's "Elements" below.

Explanatory Charts, Diagrams, Photos

Helpful Guidance

  1. 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, Geometric and/or Trigonometric Art, Charting/Diagramming and Algebraic Formulation.
    • For more art charts and graphs, you might also want to click on Microsoft Excel Imagery, Mathematics, Spreadsheets or Graphics to view many Excel worksheets and charts where Trigonometry, Geometry and Calculus have been turned into Art, or simply click on the category as appears in the upper right white portion of this page, or at the bottom left of the page.

Video assistance

Tips

  • The proof Euclid does depends upon his proof of the Pythagorean Theorem; here is a picture of that proof:

  • To help with understanding how angles with equal bases in a circle have the same angle at their far ends where they touch the circle again, two pictures of Euclid's previous theorems, BOOK III Propositions 20 and 21 are here reproduced:

  • It was stated above that Euclid's own proof, Book III P35, was much longer and more involved, in that it also includes the Pythagorean Theorem proof. Here is a picture of the proof:

Related Articles

Sources and Citations

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