A generalization of the Pythagorean theorem

G. D. Chakerian [1] discovered the following generalization of the Pythagorean theorem. Take any triangle and attach squares to each side. From each triangle vertex, extend a line perpendicular to the opposite side and extend the line through the square, as shown in the figure below. Then the areas of the rectangles with the same labels are equal. It’s not clear at first why this is a generalization of the Pythagorean theorem, but if we make our triangle closer to a right triangle it’s easier to see the connection. As the angle at B becomes closer to a right angle, the area of the orange triangles labeled II goes to zero. In the limit we’re left a theorem saying that the areas of the blue and green squares add up to the area of the square attached to the hypotenuse. The paper [1] caught my eye because it gives a connection between a geometry problem and a circuit problem, much like this post. However, I find the theorem itself more interesting than the connection to circuits. The…