Feuerbach Point

Author

Eric W. Weisstein
January 3, 2007

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

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

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

Graphic

t=Triangle[{{0,0},{1,0},{.2,.5}}];
Show[Graphics[{
t,
{Red,NinePointCircle[t]},
{Blue,Incircle[t]},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
{
{FeuerbachPoint[t],"F",Purple},
{Incenter[t],"I",Blue},
{NinePointCenter[t],"N",Red}
}
],{PointSize[.03],LabelOffset->.03}]
}],AspectRatio->Automatic,PlotRange->All]

Distances

FCl

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

ToAngles[%]

FF

Distance[FeuerbachPoint,FeuerbachPoint]

FG

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

ToAngles[%]

FullSimplify[Sqrt[%],TriangleQ[]&&IK>0&&OI>0&&Area>0]

FGe

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

ToAngles[%]

FH

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

ToAngles[%]

FI

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

ToAngles[%]

FK

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

ToAngles[%]

FL

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

ToAngles[%]

FM

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

ToAngles[%]

FN

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

ToAngles[%]

FNa

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

ToAngles[%]

FO

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

ToAngles[%]

FSp

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

ToAngles[%]

Midpoint Theorem

t=Triangle[{{0,0},{1,0},{.3,.5}}];
m=Midpoint/@Sides[t];
f=FeuerbachPoint[t];
Show[Graphics[{
t,
EqualSideTicksAboutMidpoints[t,.02,.02],
{Red,
Line[Coordinates/@{f,#}]&/@m
},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{m,{"X","Y","Z"}}],
{
{f,"F",Blue}
}
],{PointSize[.03],LabelOffset->.03}]
}],AspectRatio->Automatic,PlotRange->All]

d=Sqrt[#.#]&/@(Subtract@@@Coordinates/@(Line[Coordinates/@{f,#}]&/@m))

Chop[d[[2]]+d[[3]]-d[[1]]]