Ptolemy's Inequality is a famous inequality attributed to the Greek mathematician Ptolemy.
The inequality states that in for four points in the plane,
This also holds if are four points in space not in the same plane, but equality can't be achieved.
Proof for Coplanar Case
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 triangles and are similar, this would imply that the angles and are congruent, i.e., that is a cyclic quadrilateral.
Outline for 3-D Case
Construct a sphere passing through the points and intersecting segments and . We can now prove it through similar triangles, since the intersection of a sphere and a plane is always a circle.
Proof for All Dimensions?
Let any four points be denoted by the vectors .
From the Triangle Inequality,