Consider a
trapezoid like this:
_b__
a | ⁄
|⁄
b |\
|_\
a
Where the
trapezoid has bases of
length a and
length b , and a
height of
length a + b .
The
area of the
trapezoid can be found by the conventional formula:
½(base1 + base2) × height
= ½(a + b)(a + b)
The
area of the
trapezoid can also be found by adding the
area of the
triangles:
½ab + ½ab + ½cc
= ab + ½cc
Set these two
equal, and
solve:
ab + ½cc = ½(a + b)(a + b)
2ab + cc = (a + b)(a + b)
2ab + cc = aa + 2ab + bb
cc = aa + bb
[]
In
book 1 of
Euclid's Elements,
the pythagorean theorem appears as
proposition 47, and its
converse appears as
proposition 48.