de Longchamps Point

Author

Eric W. Weisstein
January 3, 2007

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{Green,Circumcircle[t]},
{Blue,perp=PerpendicularBisector[l=Line[Coordinates/@{DeLongchampsPoint[t],Orthocenter[t]}],R=Circumradius[t]]
},
{Red,PerpendicularBisector[perp,R],
EqualSideTicksAboutMidpoint[l,1,.025]
},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
{
{DeLongchampsPoint[t],"L",Red},
{Orthocenter[t],"H",Orange},
{Circumcenter[t],"O",Blue}
}
],{PointSize[.035],LabelOffset->.04}]
}],
AspectRatio->Automatic,PlotRange->All]

CollinearQ[{Orthocenter,Circumcenter,DeLongchampsPoint}]//Timing

Orthocenter of Anicomplementary Triangle

Show[Graphics[{
t,
{Red,
at=AnticomplementaryTriangle[t],
PerpSquare[#,.06,2]&/@
Transpose[{altat=Altitudes[at],Sides[at]}],
EqualSideTicksAboutMidpoints[at,.035,.015],
Dashing[{.02}],
altat
},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{Point/@Vertices[at],{"A'","B'","C'"}}],
{
{Orthocenter[at],"H' = L",Red}
}
],{PointSize[.03],LabelOffset->.06}]
}],AspectRatio->Automatic,PlotRange->All]

{Orthocenter[at],DeLongchampsPoint[t]}

Euler and Soddy Lines

t=Triangle[{{0,0},{1,0},{.2,.4}}];
Show[Graphics[{
t,
{Blue,EulerLine[t]},
{Green,SoddyLine[t]},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
{
{DeLongchampsPoint[t],"L",Red}
}
],{PointSize[.03],LabelOffset->.05}]
}],AspectRatio->Automatic,
PlotRange->{{-.2,1.1},{-.6,.5}}
]

Distances

LCl

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

ToAngles[%]

LF

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

ToAngles[%]

LG

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

ToAngles[%]

LGe

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

ToAngles[%]

LH

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

ToAngles[%]

LI

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

ToAngles[%209/((a^4)-(a^3) b-a (b^3)+(b^4)-(a^3) c+(a^2) b c+a (b^2) c-(b^3) c+a b (c^2)-a (c^3)-b (c^3)+(c^4))]

ToAngles[%%]

LK

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

ToAngles[%]

LL

Distance[DeLongchampsPoint,DeLongchampsPoint]

LM

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

ToAngles[%]

LN

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

ToAngles[%]

LNa

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

ToAngles[%]

LO

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

ToAngles[%]

LSp

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

ToAngles[%]