There is a large class of proofs of the Pythagorean Theorem which show that a square of side length $c$ can be dissected into squares of side lengths $a$ and $b$.

There is also the proof which shows that the right triangle itself can be dissected into similar triangles whose hypotenuses have length $a$ and $b$.

Are there any natural-looking proofs which dissect some shape other than a square or right triangle, like a dissection of an equilateral triangle of side length $c$ into equilateral triangles of side lengths $a$ and $b$? (By Bolyai's theorem we know that such a dissection exists, but it might be really painful to construct...)

  • $\begingroup$ (Compare math.stackexchange.com/questions/1571578/…, except that my second paragraph excludes the answer given.) $\endgroup$ – Micah Oct 4 '18 at 3:50
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    $\begingroup$ This may not be sufficiently "unusual", but ... U.S. President James Garfield devised a proof by dissection of a trapezoid (MAA.org). $\endgroup$ – Blue Oct 4 '18 at 4:20
  • $\begingroup$ For something insufficiently "natural-looking" ... Any proof involving dissecting the squares on each side of a right triangle converts to an equilateral triangle in the following way: Apply compatible linear transformations to the squares to turn them into $60^\circ$-$120^\circ$ rhombi. Cutting along each short diagonal gives a pair of equilateral triangles; duplicating each pair gives four triangles, which assemble into a single triangle twice as large. Scale by $1/2$ for triangles of side-length $a$, $b$, $c$. $\endgroup$ – Blue Oct 4 '18 at 5:45

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