Gergonne Point

Author

Eric W. Weisstein
January 3, 2007

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

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

©2007 Wolfram Research, Inc. except for portions noted otherwise

Figure

t=Triangle[{{0.,0.},{1,0},{.2,.7}}];
Show[Graphics[{
t,
{
Blue,
Incircle[t],
Dashing[{.02}],
ContactTriangle[t]
},
 {
Red,Dashing[{.02}],
Line/@Transpose[Vertices/@{ContactTriangle[t],t}]
},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{Point/@Vertices[ContactTriangle[t]],{"(Subscript[T, A])","(Subscript[T, B])","(Subscript[T, C])"}}],
{{GergonnePoint[t],"Ge",Red}}
],{PointSize[.02],LabelOffset->.04}
]
}],AspectRatio->Automatic,PlotRange->All]

Trilinears

CenterFunction[GergonnePoint]

-(((a+b+c))/(4(a b c)))Together[DistanceTrilinears[Sec[A/2]^2/.Sec[x_/2]:>Sqrt[2/(1+Cos[x])]]]//Factor

Distances

GeCl

Factor[FullSimplify[Together//@ToSidelengths[Distance[GergonnePoint,ClawsonPoint]^2]/.Sqrt[x_]Sqrt[y_]:>FullSimplify[Sqrt[x y],TriangleQ[]],TriangleQ[]]]

ToAngles[%]

GeF

Factor[FullSimplify[ToSidelengths[Distance[GergonnePoint,FeuerbachPoint]^2/.Cos[x_]:>TrigExpand[Cos[x]]],TriangleQ[]]]

ToAngles[%]

GeG

Factor[FullSimplify[ToSidelengths[Distance[GergonnePoint,Centroid]^2],TriangleQ[]]]

ToAngles[%]

GeGe

Distance[GergonnePoint,GergonnePoint]

GeH

Factor[FullSimplify[ToSidelengths[Distance[GergonnePoint,Orthocenter]^2],TriangleQ[]]]

ToAngles[%]

GeI

Factor[FullSimplify[ToSidelengths[Distance[GergonnePoint,Incenter]^2],TriangleQ[]]]

ToAngles[%]

GeK

Factor[FullSimplify[ToSidelengths[Distance[GergonnePoint,SymmedianPoint]^2],TriangleQ[]]]

ToAngles[%]

GeL

Factor[FullSimplify[ToSidelengths[Distance[GergonnePoint,DeLongchampsPoint]^2],TriangleQ[]]]

ToAngles[%]

GeM

Factor[FullSimplify[ToSidelengths[Distance[GergonnePoint,Mittenpunkt]^2],TriangleQ[]]]

ToAngles[%]

GeN

Factor[FullSimplify[ToSidelengths[Distance[GergonnePoint,NinePointCenter]^2/.Cos[x_]:>TrigExpand[Cos[x]]],TriangleQ[]]]

ToAngles[%]

GeNa

Factor[FullSimplify[ToSidelengths[Distance[GergonnePoint,NagelPoint]^2],TriangleQ[]]]

ToAngles[%]

GeO

Factor[FullSimplify[ToSidelengths[Distance[GergonnePoint,Circumcenter]^2],TriangleQ[]]]

ToAngles[%]

GeSp

Factor[FullSimplify[ToSidelengths[Distance[GergonnePoint,SpiekerCenter]^2],TriangleQ[]]]

ToAngles[%]

Construction

ct=TrilinearVertexMatrix[ContactTriangle]//InputForm

c (a+b-c)Intersections[{{1,0,0},ct[[1]]},{{0,1,0},ct[[2]]}]//Factor

((-a+b-c)b Intersections[{{1,0,0},ct[[1]]},{{0,0,1},ct[[3]]}])//Factor

a^2 * (a^3 - a^2*b + a*b^2 - b^3 - a^2*c + b^2*c + a*c^2 + b*c^2 -c^3)//Factor