Vecten Point

Author

Eric W. Weisstein
April 25, 2004

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

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

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

Externally Erected Square Centers

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)}}))

TrigExpand[Cos[C-Pi/4]]

Position of (Subscript[X, 485])

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)}}));

Simplify[Intersections[{m[[1]],{1,0,0}},{m[[2]],{0,1,0}}]]4c

((%)/(a b c (Cos[A]+Sin[A]) (Cos[B]+Sin[B])(Cos[C]+Sin[C])))

CenterFunction[X[485]]

Simplify[TrigExpand[CenterFunction[X[485]]]]

External Figure

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,
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]
]

Internally Erected Square Centers

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[%]

Simplify[TrigExpand[Sec[Pi/4+A]]]

Position of (Subscript[X, 486])

Simplify[Intersections[{{1,0,0},m[[1]]},{{0,1,0},m[[2]]}]]4c

Simplify[Intersections[{{1,0,0},m[[1]]},{{0,0,1},m[[3]]}]]4b

%/(-a b c(Cos[A]-Sin[A])(Cos[B]-Sin[B]) (Cos[C]-Sin[C]) )

FullSimplify[%]

Simplify[TrigExpand[Sec[Pi/4+A]]]

Inner Vecten Point

Simplify[Intersections[{{1,0,0},m[[1]]},{{0,1,0},m[[2]]}]]4c

Simplify[Intersections[{{1,0,0},m[[1]]},{{0,0,1},m[[3]]}]]4b

%/(-a b c(Cos[A]-Sin[A])(Cos[B]-Sin[B]) (Cos[C]-Sin[C]) )

FullSimplify[%]

Simplify[TrigExpand[Sec[Pi/4+A]]]

Inner Figure

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,
LongestLine/@Transpose[{Vertices[s],RotateRight[Vertices[t]],Table[Coordinates[X[486][t]]//Simplify,{3}]}]
},
PointLabels[Join[
Append[#,Red]&/@Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[s],{"(Subscript[I, C])","(Subscript[I, A])","(Subscript[I, B])"}}],
{{X[486][t],"(Subscript[X, 486])",Purple}}
],PointSize[.025]]
},AspectRatio->Automatic]
]

Collinearity of (Subscript[X, 5]), (Subscript[X, 485]), (Subscript[X, 486])

CenterName/@X/@{5,485,486}

CenterFunction/@X/@{5,485,486}

CollinearQ[X/@{5,485,486}]