How to Understand the Beautiful Crossing Chords Theorem of Euclid

Step-by-Step Guide

1. 1

   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.
   
   I will teach you in a few steps to prove how this is true.
2. 2

   Step 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 ; 2) ABC = ADC because inscribed angles subtended by the same chord AC are equal ; and 3) 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.
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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.

4. 4

   Step 4: Prove that the first identity above

5. 5

   Step 5: AP/PC = BP/PD

6. 6

   Step 6: leads directly to the Intersecting Chords Theorem

7. 7

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

8. 8

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

9. 9

   Step 9: and uses the Pythagorean Theorem

10. 10

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

Detailed Guide

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