Evans Point

Author

Eric W. Weisstein
March 28, 2003

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

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

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

Initialization

<<PlaneGeometry`;
<<KimberlingCenters`;

Figure

t=Triangle[{{0,0},{1,0},{.4,.6}}];
n=NobbsPoints[t];
Show[Graphics[{
{Blue,EulerLine[t,2]},
{Red,LongestLine[Coordinates/@n]},
t,
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{NobbsPoints[t],{"F'","D'","E'"}}],
Append[#,Blue]&/@{
{Orthocenter[t],"H"},
{Centroid[t],"G"},
{Circumcenter[t],"O"}
},
{
{EvansPoint[t],"Ev",Purple}
}
],LabelOffset->{.12,-.09},PointSize[.015]]
}],AspectRatio->Automatic,PlotRange->All
]

Trilinears

FullSimplify[((c Cos[C]-a Cos[A])(s-c)-(a Cos[A]-b Cos[B])(s-b))/a/.s->Semiperimeter[]]

Together//@DistanceTrilinears[%]

FullSimplify[%,TriangleQ[]]

CenterFunction[EvansPoint]

CenterFunction[DeLongchampsPoint]

BisymmetricQ[EvansPoint]

IncidentCenters[KimberlingCenters,EvansPoint]

Gergonne Line-Euler Line Intersection

c1=First[Intersections@@(CyclicTrilinears/@LineFunction/@{GergonneLine,EulerLine})]

c2=((-2 (a^2) Cos[A]+b (a-b+c) Cos[B]+(a+b-c) c Cos[C])/(2 a));

Together//@DistanceTrilinears[Subtract@@First/@ExactTrilinears/@CyclicTrilinears/@{c1,c2}]

{c1,c2}//InputForm