# Pythagorean Theorem from the Shoelace Formula

John Molokach also observed that the Pythagorean theorem follows from Gauss' Shoelace Formula. The formula gives the area of a polygon _{1}A_{2}...A_{n}_{j} = (x_{j}, y_{j}),

Area(A_{1}A_{2}...A_{n}) = ½|∑(x_{j}y_{j+1} - x_{j+1}y_{j})| = ½|∑x_{j}(y_{j+1} - y_{j-1})|,

where the sum is taken for j = 1, ..., n and the indices are defined cyclically: _{n+1} = x_{1}_{0} = x_{n}

For the square with vertices A_{1}(b, 0), _{2}(a+b, b),_{3}(a, a+b),_{4}(0, a),

On the other hand, if c is the hypotenuse of the right triangle with vertices

A question could be asked whether the Shoelace formula, in itself, is based on the Pythagorean theorem, causing a vicious circle in proving the latter. Eschewing this question for the time being, we present Roger Nelsen's synthetic proof without words that algebraically is equivalent to the above derivation:

In the first figure on the right, one of the darker triangles has area a²/2, the other one - b²/2. The light triangles have areas

a²/2 + b²/2 + (a + b)²/2 - 2ab/2 = a² + b².

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