A Demonstration of Pythagoras' Theorem
- Andrew Stacey
The diagram demonstrates Pythagoras' Theorem (and its generalisation to the cosine rule). This demonstration works by virtue of the fact that in Euclidean geometry, everything scales as one would expect, and that translation and rotation do not affect length.
The three cojoined triangles are all scaled versions of the original triangle. The red triangle has scale factor $c$, the blue triangle scale factor $a$, and the green triangle scale factor $b$. They fit together as shown and form an isosceles trapezium. One side has length $c^2$ and the other has length $a^2 + b^2$. The overhang is $2 a b \cos(C)$. You can move the vertices of the original triangle and see how the diagram adjusts. When $C$ is a right angle, the trapezium becomes a rectangle. When $C$ is obtuse, the $c^2$ side becomes the longest, thus showing that for an obtuse angle, $\cos(C)$ is equal to $-\cos(180 - c)$.