Claim: The area(size) of the triangle in motion never changes.
Notice that we have two kinds of motion: skewing and rotation. Clearly the area of the rotating triangle isn't changing so let's examine the skewing part. Refresh the page and observe the motion of the green triangle with its tip starting from the top of side a and tracing along side b. Now remember that:
area of a triangle = ½(base × height)
even for obtuse triangles. Also, the central triangle with sides a, b and c is right(the angle between sides a and b is 90°) so side b is perpendicular to side a and parallel to the base of the triangle in motion. Therefore the height of the triangle never changes, and neither does its area. The same goes for the second skew(post rotation) since the verticle drop is parallel to the base of the triangle by construction. So the height of the triangle remains the same throughout this skewing process as well.
QED baby
The animation shows that:
½ area of above left square = ½ area of below left rectangle
hence
area of above left square = area of below left rectangle
The same goes for the right side. Combining the results of both sides:
Given a right triangle with sides a, b and c: a2 + b2 = c2 (Pythagoras Theorem)
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