The world's simplest demonstration of the Pythagorean Theorem that, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs

 

The world's simplest 

demonstration of the Pythagorean Theorem that,

in a right triangle, 

the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs.

          So if we build a square on the two legs and another on the hypotenuse, then the sum of the areas of the first two is equal to the third. The figure is classic.

     

          Instead of squares, we can build any three similar figures on the three sides. For example semicircles, as in the first figure.

            AHA ! Then we can take the height $AD$ of the triangle and automatically similar triangles are formed on each of the three sides : $\Delta ABC$ on the side $[BC]$, $\Delta ADC$ on the side $[AC]$, $\Delta ADB$ on the side $[AB]$. These triangles are similar, having a right angle and a common acute angle. Moreover, there is an obvious relationship between the areas, which is marked on the figure. Or this is precisely the Pythagorean Theorem.

​   Remark CiP This demonstration belongs to O. Bottema. I saw it in the book 

Topics in Elementary Geometry

Springer Science+Business Media, 2008

Many other demonstrations can be found on the defunct website Cut_the_Knot        

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