Ptolemy's Inequality is a famous inequality attributed to the Greek mathematician Ptolemy.
The inequality states that in for four points in the plane,
with equality if and only if is a cyclic quadrilateral with diagonals and OR if are collinear.
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 equations 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,
Note about Higher Dimensions
Similar to the fact that that there is a line through any two points and a plane through any three points, there is a three-dimensional "solid" or 3-plane through any four points. Thus in an n-dimensional space, one can construct a 3-plane through the four points and the theorem is trivial, assuming the case has already been proven for three dimensions.