Outer Vecten Triangle

Author

Eric W. Weisstein
November 8, 2005

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

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Figure

Show[Graphics[{
{Red,
ErectPolygons[Vertices@t,4,-1]
},
t,
{Blue,s},
PointLabels[Join[
Append[#,Red]&/@Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[s],{"(Subscript[O, C])","(Subscript[O, A])","(Subscript[O, B])"}}]
],PointSize[.025]]
},AspectRatio->Automatic]
]

Perpendicularity

t=Triangle[{{0,0},{1,0},{3/10,8/10}}];
s=Triangle[ErectedPolygonCenters[Vertices@t,4,-1]];

Show[Graphics[{
{Red,ErectPolygons[Vertices@t,4,-1]},
t,
{Blue,
l1=Line[Rest[Vertices[s]]],
l2=Line[{Vertices[s][[1]],Vertices[t][[3]]}],
PerpSquare[{l1,l2},.08,2]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Transpose[{VertexPoints[s],{"(Subscript[O, C])","(Subscript[O, A])","(Subscript[O, B])"}}]
]]
},AspectRatio->Automatic]
]

PerpendicularQ[l1,l2]

Trilinear Vertex Matrix

TrilinearVertexMatrix[OuterVectenTriangle]//TraditionalForm

Area

FullSimplify[Factor[Together//@ToSidelengths[Area[OuterVectenTriangle]]/.Sqrt[x_]Sqrt[y_]:>FullSimplify[Sqrt[x y],TriangleQ[]]/.Sqrt[x_]:>FullSimplify[Sqrt[x],TriangleQ[]]],TriangleQ[]]

%/.(Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])->4Area

TraditionalForm[%]

Side Lengths

FullSimplify[Together//@ToSidelengths[SideLengths[OuterVectenTriangle]]/.Sqrt[x_]Sqrt[y_]:>FullSimplify[Sqrt[x y],TriangleQ[]]/.Sqrt[x_]:>FullSimplify[Sqrt[x],TriangleQ[]],TriangleQ[]]

TraditionalForm[%/.(Sqrt[-((a-b-c) (a+b-c) (a-b+c) (a+b+c))])->Area]

Outer Vecten Point

Show[Graphics[{
{Red,
ErectPolygons[Vertices@t,4,-1]
},
t,
{Blue,
Line/@Transpose[{Vertices[s],RotateRight[Vertices[t]]}]
},
PointLabels[Join[
Append[#,Red]&/@Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[s],{"(Subscript[O, C])","(Subscript[O, A])","(Subscript[O, B])"}}],
{{X[485][t],"(Subscript[X, 485])",Purple}}
],PointSize[.025]]
},AspectRatio->Automatic]
]

Derivation

t=Triangle[{{0,0},{1,0},{3/10,8/10}}];
s=Triangle[ErectedPolygonCenters[Vertices@t,4,-1]];

ExactTrilinears[CartesianToTrilinear[t,VertexPoints[s][[2]]]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]//Simplify

SideLengths[Evaluate[t]]

As expected, (Subscript[α, 1])= -a/2

LineEquation[{{0,1,0},{0,0,1}}]

Collect[LineEquation[{{0,c,b},{-a/2,β1,γ1}}],{α,β,γ}]

The side B C and line from the midpoint of B C to the center of the square are perpendicular, so

l1=(l ll+m mm+n nn)-(m nn+mm n)Cos[A]-(n ll+nn l)Cos[B]-(l mm+ll m)Cos[C]/.Thread[{l,m,n}->{1,0,0}]/.Thread[{ll,mm,nn}->{(-b β1+c γ1),-(1/2) a b,((a c )/2)}]//FullSimplify

soln=Simplify[Solve[{
l1==0,
Distance[{-a/2,β1,γ1},{0,c,b}]==a/2
},{β1,γ1}],c>0]

coords=FullSimplify[DistanceTrilinears[Last/@soln[[2]]],TriangleQ[]]

Area[]

coords=Prepend[Simplify[LawOfCosines[{(((a^2)+(b^2)-(c^2)+4Δ)/(4 b)),(((a^2)-(b^2)+(c^2)+4Δ)/(4 c))}]],-(a/2)]

MapIndexed[RotateRight[#1,#2[[1]]-1]&,CyclicTrilinears[coords]]//TraditionalForm

m=(({{-(a/2), (1/2) a (Sin[C]+ cos(C)), (1/2) a (Sin[B]+cos(B))}, {(1/2) b (Sin[C]+ cos(C)), -(b/2), (1/2) b (Sin[A]+cos(A))}, {(1/2) c (Sin[B]+cos(B)), (1/2) c (Sin[A]+ cos(A)), -(c/2)}}))

FullSimplify[(1/a)((a c Sin[B]+a c Cos[B]) (a b Sin[C]+a b Cos[C]))]

ps=TrilinearToCartesian[t,
#/.Δ->Area[]/.Thread[{A,B,C}->Angles[t]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]]&/@MapIndexed[RotateRight[#1,#2[[1]]-1]&,CyclicTrilinears[coords]];

Show[Graphics[{
{Red,
ErectPolygons[Vertices@t,4,-1]
},
t,
{PointSize[.04],Green,Point/@ps},
PointLabels[Join[
Append[#,Red]&/@Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[s],{"(Subscript[O, C])","(Subscript[O, A])","(Subscript[O, B])"}}]
],PointSize[.025]]
},AspectRatio->Automatic]
]

Centers

(pts={#,KimberlingLookup[#[OuterVectenTriangle]]}&/@Take[KimberlingCenters,100]);//Timing

{#[[1]],Name[#[[1]]],X[#[[2,1]]],Name[X[#[[2,1]]]]}&/@Cases[pts,_?(Last[#]=!={}&)]//TableForm