Here’s a neat theorem from geometry.
Begin with any triangle. Let R be the radius of its circumscribed circle and r be the radius of its inscribed circle. Let a, b, and c be the signed distances from the center of the circumscribed circle to the three sides. The sign of a, b, and c is negative if the segment joining the circumcenter to the side does not pass through the interior of the triangle (such as the value b shown below, represented by the teal segment), and it is positive otherwise.
Then we have the following elegant result:
Carnot’s theorem. a+b+c=R+r
Check out this GeoGebra applet that I created to see this theorem in action.
Recently I wrote about the Japanese Theorem. If you were unsuccessful in proving this beautiful theorem, try again using Carnot’s Theorem.