Brocard Diameter

Author

Eric W. Weisstein
October 25, 2004

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

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

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

Equation

LineEquation[{SymmedianPoint,Circumcenter}]

Factor/@Collect[a b c DistanceTrilinears[%]//FullSimplify,{α,β,γ}]

LineEquation[BrocardAxis]

Length

len=FullSimplify[Together//@DistanceTrilinears[Distance[Circumcenter,SymmedianPoint]],a>0&&b>0&&c>0]

Simplify[len R/Circumradius[{a,b,c}],a>0&&b>0&&c>0]

FullSimplify[With[{ω=BrocardAngle[{a,b,c}]},
R Sqrt[1-4Sin[ω]^2]/Cos[ω]
],a>0&&b>0&&c>0]

Midpoint

mid=Midpoint[{Circumcenter,SymmedianPoint}][[1]]

FullSimplify[DistanceTrilinears[mid]]

alpha=FullSimplify[TrigExpand[DistanceTrilinears[Cos[A-BrocardAngle[{a,b,c}]]]],a>0&&b>0&&c>0]

2 alpha(Sqrt[(b^2) (c^2)+(a^2) ((b^2)+(c^2))])