Weitzenböck's Inequality

Author

Eric W. Weisstein
February 19, 2003

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

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

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

Initialization

<<PlaneGeometry`;

Proof

Experimental`Resolve[ForAll[
{a,b,c},
TriangleQ[{a,b,c}],
a^2+b^2+c^2>=4Sqrt[3]TriangleArea[{a,b,c}]
]]//Timing

Experimental`Resolve[Exists[
{a,b,c},
TriangleQ[{a,b,c}],
a^2+b^2+c^2==4Sqrt[3]TriangleArea[{a,b,c}]
]]//Timing

Equality

eq=Reduce[a^2+b^2+c^2==4Sqrt[3]TriangleArea[{a,b,c}]&&TriangleQ[{a,b,c}],ReplaceValues->True]

Roots[(c^4 - 2*c^2*#1^2 + #1^4 & )[x] == 0, x]

eq //. Root[c^4 - 2*c^2*#1^2 + #1^4 & , 3] -> c