Claim: For a given side, the area of the triangle in motion never changes.
Notice that we have two kinds of motion: rotation and skew. Clearly the area of the rotating triangle isn't changing so let's examine the skew. Refresh the page observing the motion of the green triangle with its tip starting from the top of side a and tracing along side b. Now remember:
area of a triangle = ½(base × height)
even for obtuse triangles. Also, since the central triangle is right(the angle between sides a and b is 90°) side b is perpendicular to side a and parallel to the base of the triangle in motion. So the height of the triangle never changes, and neither does its area. The same goes for the second skew since the verticle drop is parallel to the base of the skewing triangle.
QED baby
The animation shows that:
½ area of above left square = ½ area of below left rectangle
which means that
area of above left square = area of below left rectangle
And the same goes for the right side. Combining both sides we get:
Given a right triangle with sides a, b and c: a2 + b2 = c2 (Pythagoras)
Full disclosure: The idea for this came to me back in the 90's after reading two books: Fractals Everywhere(fig. 2.5.1: path continuously deformed...) and Euclids Elements(Prop. 47). Jim Morey beat me to the internet with a Java applet earning him the grand prize for programming at the first Sun/Java conference in 1995. Applets are no longer supported on the web so here is an SVG/Javascript version with commentary. © 2026 Joe Hartman