1.
Proofs of Pythagorean Theorem
1 Proof by Pythagoras (ca. 570 BC–ca. 495 BC) (on the left)
and by US president James Garfield (1831–1881) (on the right)
Proof by Pythagoras: in the figure on the left, the area of the large square (which is equal to
(a + b)2 ) is equal to the sum of the areas of the four triangles ( 21 ab each triangle) and the area of
the small square (c2 ):
2 1
(a + b) = 4 ab + c2 ⇒ a2 + 2ab + b2 = 2ab + c2 ⇒ a2 + b2 = c2 .
2
2 Proof by Bhaskara (1114–1185)
1
2.
3 Proof by similar triangles
Let CH be the perpendicular from C to the side AB in the right triangle 4ABC.
AB CB c a a2
Observation 1: 4ABC ∼ 4CBH, therefore = , i.e., = , hence e = . (1)
BC CH a e c
AB AC c b b2
Observation 2: 4ABC ∼ 4ACH, therefore = , i.e., = , hence d = . (2)
AC AH b d c
a2 b2
Finally, AB = BH + AH, i.e, c = e + d. Using (1) and (2), we rewrite this as c = + , which is
c c
equivalent to c2 = a2 + b2 .
The book
Elisha Scott Loomis, The Pythagorean Proposition: Its Demonstrations Analyzed and
Classified, and Bibliography of Sources for Data of the Four Kinds of “Proofs”, Second
edition, 1940, available at http://files.eric.ed.gov/fulltext/ED037335.pdf
contains 370 proofs of the Pythagorean Theorem.
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