Editing Quadrilateral (section)

===Diagonals and bimedians===
A corollary to Euler's quadrilateral theorem is the inequality
:<math> a^2 + b^2 + c^2 + d^2 \ge p^2 + q^2 </math>

where equality holds if and only if the quadrilateral is a [[parallelogram]].

[[Leonhard Euler|Euler]] also generalized [[Ptolemy's theorem]], which is an equality in a [[cyclic quadrilateral]], into an inequality for a convex quadrilateral. It states that
:<math> pq \le ac + bd </math>

where there is equality [[if and only if]] the quadrilateral is cyclic.<ref name=Altshiller-Court/>{{rp|p.128–129}} This is often called [[Ptolemy's inequality]].

In any convex quadrilateral the bimedians ''m, n'' and the diagonals ''p, q'' are related by the inequality
:<math>pq \leq m^2+n^2,</math>

with equality holding if and only if the diagonals are equal.<ref name=J2014>{{cite journal |last=Josefsson |first=Martin |title=Properties of equidiagonal quadrilaterals |journal=Forum Geometricorum |volume=14 |year=2014 |pages=129–144 |url=http://forumgeom.fau.edu/FG2014volume14/FG201412index.html |access-date=2014-08-28 |archive-date=2024-06-05 |archive-url=https://web.archive.org/web/20240605032351/https://forumgeom.fau.edu/FG2014volume14/FG201412index.html |url-status=dead }}</ref>{{rp|Prop.1}} This follows directly from the quadrilateral identity <math>m^2+n^2=\tfrac{1}{2}(p^2+q^2).</math>