Ptolemy's Inequality
Ptolemy's Inequality states that in for four points in the plane,
,
with equality iff. is a cyclic quadrilateral with diagonals and .
Proof
We construct a point such that the triangles are similar and have the same orientation. In particular, this means that
.
But since this is a spiral similarity, we also know that the triangles are also similar, which implies that
.
Now, by the triangle inequality, we have . Multiplying both sides of the inequality by and using and gives us
,
which is the desired inequality. Equality holds iff. , , and are collinear. But since the angles and are congruent, this would imply that the angles and are congruent, i.e., that is a cyclic quadrilateral.