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The 3-4-5 right triangle and its squares
a² b² c² a = 3 b = 4 c = 5 placeholder

a² + b² = c². For the 3-4-5 triangle: 9 + 16 = 25. The blue and red squares together equal the green square in area.

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Pythagorean triples: integer solutions to a² + b² = c²

Table of Pythagorean triples

abca²+b²=c²
3459+16=25 ✓
5121325+144=169 ✓
8151764+225=289 ✓
7242549+576=625 ✓
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Visual proof: the same four triangles, rearranged
c² Same 4 triangles ⟹ b² a² Same 4 triangles c² = a² + b² Both arrangements have the same total area.

Both big squares are (a+b)×(a+b). Both contain four identical right triangles. What is left over in the left square is c². What is left over in the right square is a²+b². They must be equal.

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