Colburn's Proof of the Pythagorean Theorem

The proof of the Pythagorean Theorem illustrated by the applet below appears as proof One Hundred One in E. S. Loomis' collection. Loomis credits the proof to A. R. Colburn who published 91 proofs in the Scientific American Supplement in 1917.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

What if applet does not run?

Let ΔABC be a triangle with a right angle at C. As usual, hypotenuse AB = c, BC = a, AC = b. Find x, y, z such that:

a = y + z
b = x + y
c = x + y + z.

(The unique construction should be clear from the applet.) The Pythagorean Theorem is then equivalent to the assertion that

(1)	(y + z)² + (x + y)² = (x + y + z)²,

which easily reduces to

(2)

y² = 2xz.

However, (2) also appears in the calculations of the power of point A with respect to circle B(C), i.e. the circle centered at B and passing through C. Indeed, the power of point A can be evaluated in two ways:

(3)	(x + y)² = x(x + 2(y + z)),

which is equivalent to (2).

References

E. S. Loomis, The Pythagorean Proposition, NCTM, 1968

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