Proof of Pythagoras

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Most people can recite Pythagoras’ Theorem: The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides. Very few of us can conjure up in our minds the concrete reasons that it is true.

This illustration dates back to the time of the great Greek mathematician Euclid in the 4th century BC and relies on a couple of geometric truths:

  • Triangles are described in terms of the length of the sides and the size of the angles. If you know any combination of three things taken from those, you have enough to completely define the triangle.
  • Any two triangles which share the same base and have the same height, even if the shapes look very different, have the same area.
  • A triangle whose base is one side of a rectangle and height height is the same as an adjacent side of the rectangle is exactly half the area of the rectangle.

The proof works by dividing the square based on the hypotenuse – the longest side of the central triangle – into two rectangles.

  • Each of the two non-hypotenuse sides then has two triangles based on it.
  • Each of the two smaller triangles based on one side of the main triangle shares a common angle and two common sides with one of the smaller triangles based on the other side of the man triangle – so they are identical.
  • One of the two identical triangles is half the area of the square built on that side and the other is half the area of one of the rectangles into which the largest square is split Since the triangles are the same size, the rectangle and square must be the same size.
  • The two different rectangles being the same size as the two different squares, the large square (which is just the two rectangles added together) must be the same size as the two squares added together.

This illustration breaks it down visually…

Pythagoras-from-Euclid-3

And here are the colour variants of the design…

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