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How to Understand the Beautiful Crossing Chords Theorem of Euclid

Steps

  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. I will teach you in a few steps to prove how this is true.
  2. Prove the similarity of triangles ABP and CDP that is a consequence of their angles since
    • 1) BAD = BCD because inscribed angles subtended by the same chord BD are equal [Book III Propositions 20 and 21];
    • 2) ABC = ADC because inscribed angles subtended by the same chord AC are equal [Book III Propositions 20 and 21]; and
    • 3) 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.

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  • 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 was much longer and more involved, in that it also includes the Pythagorean Theorem proof.





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Date: 07.12.2018, 22:01 / Views: 95385