Pythagorean theorem three sides in place of squares (squares are a special case, of course). The upper figure shows that for a scalene triangle, the area of the parallelogram on the longest side is the sum of the areas of the parallelograms on the other two sides, provided the parallelogram on the long side is constructed as indicated (the dimensions labeled with arrows are the same, and determine the sides of the bottom parallelogram). This replacement of squares with parallelograms bears a clear resemblance to the original Pythagoras's theorem, and was considered a generalization by Pappus of Alexandria in 4 AD The
Viviani's theorem Viviani's theorem Viviani's theorem, named after Vincenzo Viviani, states that the sum of the distances from "any" interior point to the sides of an equilateral triangle equals the length of the triangle's altitude. This proof depends on the readily-proved proposition that the area of a triangle is half its base times its height—that is, half the product of one side with the altitude from that side. Let ABC be an equilateral triangle whose height is "h" and whose side is "a". Let P be any point inside the triangle, and "u, s, t" the distances of P from the sides.