Hans Walser, [20161014]

Generalization of Pythagoras

1     Problem

ŇIn the triangle ABC let any parallelograms ABED, BCFG be drawn on AB, AC and let DE, FG meet in H. Join HB and produce it to meet AC in K. The sum of the parallelograms ABED, BCFG can then be shown to the equal to the parallelogram contained by AC, HB in an angle equal to the sum of the angles BAC, DHB.Ó [1, Book IV]

Fig. 0: Red = Green + Blue

2     Proof without words

(using the principle of Cavalieri)

Fig. 1: First step

Fig. 2: Second step

Fig. 3: Third step


[1] http://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/pappus-alexandria