Euler Inequality

Author

Eric W. Weisstein
February 14, 2003

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

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

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

Initialization

<<PlaneGeometry`;

Proof

TriangleQ[{a,b,c}]

R=Circumradius[{a,b,c}]

r=Inradius[{a,b,c}]

Developer`InequalityInstance[
R<2r&&TriangleQ[{a,b,c}],
{a,b,c}]//Timing

Developer`InequalityInstance[
R==2r&&TriangleQ[{a,b,c}],
{a,b,c}]//Timing

Equality Holds

Reduce[R==2r&&TriangleQ[{a,b,c}],ReplaceValues->True]

(c^2) #1-2 c ((#1)^2)+((#1)^3)&[x]

Roots[c^2*x - 2*c*x^2 + x^3 == 0, x]