Ptolemy Inequality

Author

Eric W. Weisstein
February 20, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/PtolemyInequality.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/PtolemyInequality.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Proof

AB\[Cross]CD+BC\[Cross]DA>AC\[Cross]BD

ConcyclicQ[Quadrilateral[Transpose[Array[#,4]&/@{x,y}]]]

Resolve[ForAll[
Evaluate[Flatten[Array[#,4]&/@{x,y}]],
4 ((((x[1]-x[2]))^2)+(((y[1]-y[2]))^2)) ((((x[2]-x[3]))^2)+(((y[2]-y[3]))^2)) ((((x[3]-x[4]))^2)+(((y[3]-y[4]))^2)) ((((-x[1]+x[4]))^2)+(((-y[1]+y[4]))^2))>=(((((((x[1]-x[3]))^2)+(((y[1]-y[3]))^2)) ((((x[2]-x[4]))^2)+(((y[2]-y[4]))^2))-((((x[1]-x[2]))^2)+(((y[1]-y[2]))^2)) ((((x[3]-x[4]))^2)+(((y[3]-y[4]))^2))-((((x[2]-x[3]))^2)+(((y[2]-y[3]))^2)) ((((-x[1]+x[4]))^2)+(((-y[1]+y[4]))^2))))^2)
],Reals]//Timing