Symmedian Point

Author

Eric W. Weisstein
January 3, 2007

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

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

©2006 Wolfram Research, Inc. except for portions noted otherwise

Comments

Redo the figures

Distances

KCl

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

ToAngles[%]

KF

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

ToAngles[%]

KG

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

ToAngles[%]

KGe

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

ToAngles[%]

KH

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

ToAngles[-Denominator[%]]

ToAngles[%%]

KI

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

ToAngles[%]

KK

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

KL

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

ToAngles[%]

KM

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

ToAngles[%]

KN

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

ToAngles[%]

KNa

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

ToAngles[%]

KO

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

ToAngles[-Denominator[%]]

ToAngles[%%]

KSp

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

ToAngles[%]

Graphics

tri={{0.,0.},{1.2,1.2},{3.,0.}};
med[i_][tri_]:=MedialTriangle[tri][[1,i]];
sym[i_][tri_]:=SymmedialTriangle[tri][[1,i]];
pts={
	med[1],"(Subscript[M, A])",	med[2],"(Subscript[M, B])",	med[3],"(Subscript[M, C])",
		tri[[1]],"A",tri[[2]],"B",tri[[3]],"C",
		Centroid,"G"
};
pts2={
		LemoinePoint,"K",
			sym[1],"(Subscript[K, A])",	sym[2],"(Subscript[K, B])",	sym[3],"(Subscript[K, C])"
};
Show[Graphics[{
			Triangle[tri],
			Medians[tri],Line[Broc
			{Dashing[{.02,.02}],AngleBisectors[tri]},
			{GrayLevel[.5],Symmedians[tri]},
			PointLabels[tri,pts],
			{GrayLevel[.5],PointLabels[tri,pts2]}
			}],AspectRatio->Automatic]

tri={{0,0},{.4,1.5},{2.,0}};
m=Midpoints[tri]/.Point->Identity;
a=Altitudes[tri]/.Line->Identity;
ma=Table[Midpoint[a[[i]]],{i,3}] /. Point->Identity;
pts={
	ma[[1]],"(Subscript[A, A])",	ma[[2]],"(Subscript[A, B])",	ma[[3]],"(Subscript[A, C])",
	m[[1]],"(Subscript[M, A])",	m[[2]],"(Subscript[M, B])",	m[[3]],"(Subscript[M, C])",
		tri[[1]],"A",tri[[2]],"B",tri[[3]],"C",
		SymmedianPoint,"K"
	};

tri2={{0,0},{0,1.5},{2.,0.0001}};
m2=Midpoints[tri2]/.Point->Identity;
a2=Altitudes[tri2]/.Line->Identity;
ma2=Table[Midpoint[a2[[i]]],{i,3}] /. Point->Identity;
pts2={
		tri2[[1]],"A",tri2[[2]],"B",tri2[[3]],"C",
		SymmedianPoint,"K"
	};

Show[GraphicsArray[{
			Show[Graphics[{
			Triangle[tri],
			Table[Line[{ma[[i]],m[[addn[i,1,3]]]}],{i,3}],
			PointLabels[tri,pts],
			{Dashing[{.02,.02}],Altitudes[tri]}
		}],AspectRatio->Automatic,DisplayFunction->Identity],
Show[Graphics[{
			Triangle[tri2],
			PointLabels[tri2,pts2],
			{Dashing[{.02,.02}],Altitudes[tri2]}
		}],AspectRatio->Automatic,DisplayFunction->Identity]
		}]]

tri={{0,0},{.4,1.5},{2.,0}};
c=Circumcircle[tri];
k=SymmedianPoint[tri][[1]];
op=Table[CircleLineIntersection[{tri[[i]],k},{Circumcenter[tri][[1]],Circumradius[tri]}],
		{i,3}];
tri2={op[[1,1]],op[[2,2]],op[[3,1]]};
pts={SymmedianPoint,"K",SymmedianPoint[tri2],"K'",
		Circumcenter,"O",
		op[[1,2]],"A",op[[1,1]],"A'",
		op[[2,1]],"B",op[[2,2]],"B'",op[[3,2]],"C",op[[3,1]],"C'"};

Show[Graphics[{
			Triangle[tri],Triangle[tri2],
			Circumcircle[tri],
			PointLabels[tri,pts],
			{Dashing[{.02,.02}],Table[Line[{tri[[i]],tri2[[i]]}],{i,3}]}
}],AspectRatio->Automatic]

tri={{0.,0.},{.2,.6},{1.,0.}};
pts={SymmedianPoint,"K",SymmedialTriangle[tri][[1,2]],"(Subscript[K, A])",
			tri[[2]],"A",tri[[3]],"C",tri[[1]],"B"};
Show[Graphics[{
	Triangle[tri],
			Symmedians[tri][[2]],
			PointLabels[tri,pts,.02,{.02,.04}]
}], AspectRatio->Automatic]

Of Named Triangles

SymmedianPoint[Triangle[{{p1,q1,r1},{p2,q2,r2},{p3,q3,r3}}]]//Simplify

t=RandomTriangle[20];

rules=Join[Thread[{a,b,c}->SideLengths[Evaluate[t]]],Thread[{A,B,C}->Angles[Evaluate[t]]]];
kc=KimberlingCenters;

Length[k=Rest[#]/First[#]&/@((Trilinears/@KimberlingCenters)/.TriangleRules/.rules)]//Timing

trips=Triangles;

search::zero="Bad zero encountered `1`";
l={};
Do[
Print[l0={
i,
trips[[i]],
rc=SymmedianPoint[trips[[i]]]/.TriangleRules/.rules;
(
pt=Switch[rc,
_?(#=={0,0,0}&),{0,0,0},
Or@@Thread[#==0]&,Message[search:zero,#];rc,
_,Rest[rc]/First[rc]
];
If[pt=={0,0,0},{0},
If[#=={},{},kc[[#]]]&[Flatten[Position[k,_?(#==pt&)]]]
]
)//Timing
}];
AppendTo[l,l0],
{i,Length[trips]}
]

Table

{Name[#[[2]]],#[[-1,-1,1]],Name[#[[-1,-1,1]]]}&/@DeleteCases[l,_?(#[[-1,-1]]=={}&)]//TableForm

Structures Passing through It

pt=SymmedianPoint;

Circles

First/@Select[{#,IncidentCenters[{pt},#]}&/@Circles,Last[#]=!={}&]

Circle Centers

First/@Select[{#,IncidentCenters[{pt},CircleCenter[#]]}&/@Circles,Last[#]=!={}&]

Conics

First/@Select[{#,IncidentCenters[{pt},#]}&/@Conics,Last[#]=!={}&]

Cubics

First/@Select[{#,IncidentCenters[{pt},#]}&/@TriangleCubics,Last[#]=!={}&]

Lines

First/@Select[{#,IncidentCenters[{pt},#]}&/@Lines,Last[#]=!={}&]