Ptolemy's Inequality
From AoPSWiki
Ptolemy's Inequality is a famous inequality attributed to the Greek mathematician Ptolemy.
Theorem
The inequality states that in for four points
in the plane,
,
with equality if and only if
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.



