Mittenpunkt

Author

Eric W. Weisstein
January 3, 2007

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

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

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

Plot

t=Triangle[{{0,0},{.5,.7},{1,0}}];
j=Excenters[t];
Show[Graphics[{
t,
{
Blue,
ExcentralTriangle[t],
Excircles[t]
},
{Red,Line[Coordinates/@{Mittenpunkt[t],#}]&/@j},
PointLabels[Transpose[{Flatten[{Midpoints[t],Mittenpunkt[t]}],{"(Subscript[M, A])","(Subscript[M, B])","(Subscript[M, C])","M"}}],{Red,PointSize[.03]}],
PointLabels[Transpose[{Excenters[t],{"(Subscript[J, A])","(Subscript[J, B])","(Subscript[J, C])"}}],{PointSize[.03],Blue}],
PointLabels[Transpose[{Point/@Vertices[t],{"A","B","C"}}],PointSize[.03]]
},AspectRatio->Automatic,PlotRange->All]
]

Collinear

With Ge and G

CollinearQ[{GergonnePoint,Centroid,Mittenpunkt}]//Timing

t=Triangle[{{0,0},{.3,.3},{1,0}}];
g1=Show[Graphics[{
t,
PointLabels[Transpose[{Point/@Vertices[t],{"A","B","C"}}],{PointSize[.03],LabelOffset->{1,1}.03}],
PointLabels[{{GergonnePoint[t],"Ge"},{Centroid[t],"G"},{Mittenpunkt[t],"M"}},{Red,PointSize[.03],LabelOffset->{1,1}.03}],
{Red,LongestLine[Coordinates/@Through[{GergonnePoint,Centroid,Mittenpunkt}[t]]],
EqualSideTicks[Coordinates/@Through[{Centroid,Mittenpunkt}[t]],1,.015],
EqualSideTicks[Coordinates/@Through[{Centroid,GergonnePoint}[t]],2,.015,.07]
}
},AspectRatio->Automatic,PlotRange->All]]

Divide@@(Sqrt[#.#]&/@Subtract@@@Partition[Coordinates/@Through[{GergonnePoint,Centroid,Mittenpunkt}[t]],2,1])

With Sp and H

CollinearQ[{SpiekerCenter,Orthocenter,Mittenpunkt}]//Timing

t=Triangle[{{0,0},{.3,.3},{1,0}}];
g2=Show[Graphics[{
t,
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
Append[#,Red]&/@{{SpiekerCenter[t],"Sp"},{Orthocenter[t],"H"},
{Mittenpunkt[t],"M"}}
],{PointSize[.03],LabelOffset->{1,1}.03}],
{Red,LongestLine[Coordinates/@Through[{SpiekerCenter,Orthocenter,Mittenpunkt}[t]]]
}
},AspectRatio->Automatic,PlotRange->All]]

With I and K

CollinearQ[{Incenter,SymmedianPoint,Mittenpunkt}]//Timing

t=Triangle[{{0,0},{.3,.3},{1,0}}];
g3=Show[Graphics[{
t,
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
Append[#,Red]&/@{{Incenter[t],"I"},{SymmedianPoint[t],"K"},
{Mittenpunkt[t],"M"}}
],{PointSize[.03],LabelOffset->{1,1}.03}],
{Red,LongestLine[Coordinates/@Through[{Incenter,SymmedianPoint,Mittenpunkt}[t]]]
}
},AspectRatio->Automatic,PlotRange->All]]

Distances

MCl

Factor[FullSimplify[Together//@ToSidelengths[Distance[Mittenpunkt,ClawsonPoint]^2],TriangleQ[]]]

ToAngles[%]

MF

Factor[FullSimplify[ToSidelengths[TrigExpand[Distance[Mittenpunkt,FeuerbachPoint]^2]],TriangleQ[]]]

ToAngles[%]

MG

Factor[FullSimplify[ToSidelengths[TrigExpand[Distance[Mittenpunkt,Centroid]^2]],TriangleQ[]]]

ToAngles[%]

MGe

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

ToAngles[%]

MH

Factor[FullSimplify[ToSidelengths[TrigExpand[Distance[Mittenpunkt,Orthocenter]^2]],TriangleQ[]]]

ToAngles[%]

MI

Factor[FullSimplify[ToSidelengths[TrigExpand[Distance[Mittenpunkt,Incenter]^2]],TriangleQ[]]]

ToAngles[%]

MK

Factor[FullSimplify[ToSidelengths[TrigExpand[Distance[Mittenpunkt,SymmedianPoint]^2]],TriangleQ[]]]

ToAngles[%]

ML

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

ToAngles[%]

MM

Distance[Mittenpunkt,Mittenpunkt]

MN

Factor[FullSimplify[ToSidelengths[TrigExpand[Distance[Mittenpunkt,NinePointCenter]^2]],TriangleQ[]]]

ToAngles[%]

MNa

Factor[FullSimplify[ToSidelengths[TrigExpand[Distance[Mittenpunkt,NagelPoint]^2]],TriangleQ[]]]

ToAngles[%]

MO

Factor[FullSimplify[ToSidelengths[TrigExpand[Distance[Mittenpunkt,Circumcenter]^2]],TriangleQ[]]]

ToAngles[%]

MSp

Factor[FullSimplify[ToSidelengths[TrigExpand[Distance[Mittenpunkt,SpiekerCenter]^2]],TriangleQ[]]]

ToAngles[%]