Pythagoras: the proof of the Pythagorean Theorem

This Proof of the Pythagorean Theorem page and the entire website are excerpted from You and the Universe, a handmade, individualized fine art book on astrology, mythology and astronomy through which the recipient's complete astrological reading is woven.

Although the proposition now bearing Pythagoras' name was discovered on a Babylonian tablet (c. 1900 - 1600 BC) 1000 years before Pythagoras' time (and Pythagoras was captured by the Persians and taken captive to Babylon in 525 BC, where he spent five years before miraculously escaping to Samos), he is the first credited with actually having proved it using the deductive proof sequence of geometry. The Pythagorean Theorem states that a square drawn on hypotenuse (the longest side) of a right triangle is equal in area to the sum of the squares drawn on the other two sides (the legs). This is shown geometrically and algebraically (c² = a² + b²) in the above figure. One of many proofs now follows:

Statement	Reason
1. draw lines segments BF and AK	1. two points determine a unique line
2. through C construct CL parallel to BD intersecting AB at M	2. thru an external pt. there is 1 parallel
3. angle BAC = angle BAC	3. self-identity
4. angle FAC = angle BAE	4. all right angles are =
5. angle FAC + angle CAB = angle CAB + angle BAE	5. if =s are added to =s, the sums are =
6. ΔABF = ΔAEC	6. SAS (congruence by side-angle-side)
7. area ΔABF with altitude AC on base AF = ½ area ϨACGF	7. area Δ = ½ area Ϩ same base & alt.
8. area ΔAEC with altitude AM on base AE = ½ area AELM	8. area Δ = ½ area Ϩ same base & alt.
9. ½ area ϨACGF = ½ area ϨAELM	9. transitive property of =
10. area ϨACGF = area ϨAELM	10. if =s multiplied by =s, products are =
11. similarly area ϨBCHK = area ϨBDLM	11. counterparts of reasons 3. - 10.
12. but area ϨAELM + area ϨBDLM = area ϨABDE	12. the whole = the sum of its parts
13. area ϨABDE = area ϨACGF + area ϨBCHK Q.E.D.	13. =s may be substituted for =s
This proof is probably the most famous of the many known proofs of the Pythagorean Theorem, the figure above on which the proof draws being known as The Bride's Chair.

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The author, his poetry and instruments		Virgo and Venus in "You and the Universe"

Pythagoras and the Pythagorean Theorem.