Pythagorean Theorem by Homothetic Copies

This one was sent to me by Geoffrey Margrave from Lucent Technologies. It looks very much as #8, but is arrived at in a different way. Create 3 scaled copies of the triangle with sides a, b, c by multiplying it by a, b, and c in turn. Put together, the three similar triangles thus obtained to form a rectangle whose upper side is a² + b², whereas the lower side is c². (Which also shows that #8 might have been concluded in a shorter way.)

Also, picking just two triangles leads to a variant of Proofs #6 and #19:

In this form the proof appears in [Birkhoff, p. 92].

Yet another variant that could be related to #8 has been sent by James F.:

The latter has a twin with a and b swapping their roles.

|Pythagorean Theorem| |Contact| |Front page| |Contents| |Geometry| |Store|

Copyright © 1996-2017 Alexander Bogomolny


Search by google: