3

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...)

FD_bfa
  • 3,989
Micah
  • 38,108
  • 15
  • 85
  • 133
  • (Compare https://math.stackexchange.com/questions/1571578/jigsaw-style-proofs-of-the-pythagorean-theorem-with-non-square-squares, except that my second paragraph excludes the answer given.) – Micah Oct 04 '18 at 03:50
  • 2
    This may not be sufficiently "unusual", but ... U.S. President James Garfield devised a proof by dissection of a trapezoid (MAA.org). – Blue Oct 04 '18 at 04:20
  • 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 04 '18 at 05:45

0 Answers0