Pythagoras: the proof of the Pythagorean Theorem

The proposition now bearing Pythagoras' name was discovered on a Babylonian tablet (c. 1900-1600 BC) one thousand years before Pythagoras' time. In addition, Pythagoras was captured by the Persians and taken captive to Babylon in 525 BC, where he spent five years before miraculously escaping to Samos. Pythagoras, however, 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 both geometrically and algebraically (c² = a² + b²) in the figure above. One of many proofs follows below:

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 ■ACGF	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 ■ACGF = ½ area ■AELM	9. transitive property of =
10. area ■ACGF = area ■AELM	10. if =s multiplied by =s, products are =
11. similarly area ■BCHK = area ■BDLM	11. counterparts of reasons 3. - 10.
12. but area ■AELM + area ■BDLM = area ■ABDE	12. the whole = the sum of its parts
13. area ■ABDE = area ■ACGF + area ■BCHK 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. This Proof of the Pythagorean Theorem page and much of this 550-page resource website are excerpted from You and the Universe.

Pythagoras and the Pythagorean Theorem.