This morning I wrote a post entitled Arithmetic, Geometry, Harmony, and Gold about a theorem that relates the arithemetic mean, geometric mean, harmonic mean, and the golden ratio.
Pick two positive numbers, a and b, and label the sides of a triangle with the arithmetic, geometric, and harmonic means of a and b. When is that triangle a right triangle? There is a unique solution, up to scaling.
I also posted on @AlgebraFact a right triangle whose sides are consecutive binomial coefficients. This triangle is also unique.