# Are there nice dissection proofs of the Pythagorean Theorem which dissect something unusual?

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...)

• (Compare math.stackexchange.com/questions/1571578/…, except that my second paragraph excludes the answer given.) – Micah Oct 4 '18 at 3:50
• This may not be sufficiently "unusual", but ... U.S. President James Garfield devised a proof by dissection of a trapezoid (MAA.org). – Blue Oct 4 '18 at 4:20
• 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 $a$, $b$, $c$. – Blue Oct 4 '18 at 5:45