Six Segment Theorem
Consider four points in the plane, three of which form a triangle. The lengths of the three triangle sides, as well as the lengths of the three segments connecting the triangle vertices to the fourth point, are related as below.
You can drag around any of the points in the diagram.
This theorem, known to Japanese mathematicians as rokushajutsu (六斜術), can be proven using the Law of Cosines. An interesting special case is when x = y = z, which makes the red point coincide with triangle's circumcenter. Substituting R for each instance of those variables in the algebra above and solving, one obtains a formula for the circumradius of a triangle in terms of its side lengths:
The six segment theorem can also be applied for an efficient proof of Descartes' Circle Theorem. For more information, see Notes on the Six-Segment Theorem by J. Marshall Unger.