Roger Penrose in his book “The Road to Reality” gives a remarkably simple proof of Pythagoras’ Theorem:

Drawing a perpendicular to the hypotenuse from the right angle (line CD) will divide any right angled triangle into two similar triangles, both of which are similar to the original triangle. Since the area of similar shapes are proportional to the square of the ratio of the length of their sides, and the sum of the areas of the two smaller triangles is clearly equal to the area of the enclosing triangle, the area of the square on the long hypotenuse is equal to the sum of the squares on the other two sides (i.e. the hypotenuse of the the two smaller triangles).

The beauty of this proof is not only its simplicity, but also that it makes clear why the Pythagoras formula will only work with a right angled triangle; since only a right angled triangle can be divided into two triangles similar to the original.

This result may also be simply shown algebraically:

From similarity:

b/d = c/b

b^2 = cd

and

a/(c-d) =c/a

a^2 = c^2-cd

c^2 = a^2 + cd

hence substituting b^2 for cd:

c^2 = a^2 + b^2

A different approach to this method is given at: Terence Tao

And for those looking for some variety 87 different proofs can be found at: Cut the knot

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