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Imagine a flat, right angled triangle on a plane. "Pull up" one of the not-right-angle vertices, and join it to the other two vertices. (In other words, pick a point on a perpendicular to the plane at a not-right-angle vertex.)
We cannot have three equal triangles. If we note down the sides of the four triangles, each value must appear an even number of times -- since each side is shared by two triangles. If we had three (p,q,r) triangles, the fourth triangle would also have the same dimensions, so there would be an even number of p's, q's and r's.
We can, however, get two pairs of equal faces. If the triangle on the plane is (p,q,r) with p²+q²=r², and the distance from the "pulled up" point to the plane is also p, we'll have two (p,q,r) triangles, and two (p,r,s) ones, with p²+r²=s². |
