The Garden of Archimedes  A Museum for Mathematics

Pythagoras and his Theorem

Similar figures .

	In the enunciation of the theorem of Pythagoras, the squares can be replaced by other figures, like for example triangles, hexagons or even irregular figures, as long as they are similar among themselves.
	Similar figures are the ones that differ only in quantity, but not in shape. In other words, two similar figures are those where one is the enlargement of the other. For example, two five pointed stars are similar, while a five pointed star is not similar to a four pointed star. Arguing from analogy, two triangles in which the sides of one are twice the sides of the other are similar, while the ones in the figure are not, because in this case the enlargement is done in only one direction. Note that all squares are similar, like all circles and all regular polygons with the same number of sides. A property of similar figures that explains why they can replace the squares in the theorem of Pythagoras, is that their areas are proportional to the squares of the corresponding segments. For example, in the case of five pointed stars, the area is proportional to the square of the distance between two consecutive points; in formulas A = kL2 (Naturally, side l of the star could have been taken instead of the distance between the two points; in this case it would be A =hl 2 with a constant h different from k)
	If we now take a right-angled triangle, and adapt three stars to its three sides, as in the figure, the area of the star on the hypotenuse is equal to the sum of the areas on the cathets. In fact, for the theorem of Pythagoras we have a2+b2=c2, and multiplying by k, we'll have ka2+kb2=kc2. But because of what we just stated, the quantities ka2, kb2 and kc2 are the areas of the three stars, and therefore the area of the star on the hypotenuse is equal to the sum of the areas of those on the cathets.
	The previous observation is at the basis of an elegant demonstration of the theorem of Pythagoras. Just note that the triangles ABC and ADB in the figure are similar, since they are right-angled and have the angle in A in common. For the same reason, the triangles ABC and BDC are similar and therefore the three triangles are similar figures. Because the triangle ABC is made of the other two, its area is the sum of the areas of the other two; area (ABC)= area (ABD) + area (BDC). Since the triangles are similar, their areas are proportional to the squares of their hypotenuses, and thus k AC2 = k AB2 + k BC2 and dividing by k we get AC2 =AB2 + BC2 that is the theorem of Pythagoras.
	Lunes An interesting case is when the similar figures are semicircles. Once again the sum of the semicircles on the cathets is equal to the semicircle on the hypotenuse. If we now tilt this last one, and take away the red parts in common from the bigger semicircle and from the two small ones, the remaining figures, which are the triangle and the two yellow figures with the shape of half moons (called lunes from the Latin lunulae, small moons), will have the same area. In the case of a isosceles triangle, a lune is equal to a half triangle. This is the first case historically verified (the demonstration was attributed to Hippocrates of Chios) where it is demonstrated that a rectilinear figure (the triangle) is the same as a curved one (the lune).