# Q.E.D.

48 49 Ptolemys theorem says that for a quadrilateral inscribed in a circle a a b b d d. This reduces to Pythagoras theorem if the quadrilateral is a rectangle. Many hundred different proofs for Pythagoras theorem see page 4 have been discovered. Collected in this appendix are some of the most ingenious ones. aPPendIX I one theorem, many ProoFs Leonardo Da Vinci 14521519 noticed that the shaded areas in the two diagrams on the left have the same area and each contain two copies of the rightangled triangle. Discarding the triangles gives the theorem. A similar proof based on the diagram on the right small square 4.triangle big square b a2 4.12 ab c2 b2 2ab a2 2ab c2 a2 b2 c2 Our first proof left is based on the same diagram as the one on page 5 big square small square 4.triangle a b2 c2 4.12 ab a2 b2 2ab c2 2ab a2 b2 c2 Triangles ABC, CBD and ACD are similar. Thus, DBBC BCAB and ADAC ACAB, or BC2 AB . DB and AC2 AB .AD. Sum up AC2 BC2 AB . DB AD AB2 A proof by shearing A proof by dissection Three more proofs by dissection