Law of Cotangents
One of those ridiculously beuatiful geometry facts that you wonder how you never knew. As often, saw it at the Futility Closet, http://www.futilitycloset.com/2015/12/02/three-sides/.
Inscribe an equilateral triangle in a rectangle, or rectiscribe a rectangle around an equilateral triangle so that they share a vertex. Then the other triangles in the rectangle have a beautiful relationship. Originally from Tom Apostol and Mamikon Mnatsaka.
But wait! This is just a special case of what can only be called the Law of Cotangents. Click the checkbox and you won't believe what they said next!