Mixtilinear Triangle

Author

Eric W. Weisstein
November 8, 2005

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

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

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

Graphic

t=Triangle[{{0,0},{1,0},{.2,.5}}];
Show[Graphics[{
t,
Circumcircle[t],
Transpose[{{Red,Blue,Green},MixtilinearIncircles[t]}],
{Purple,MixtilinearTriangle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Transpose[{Center/@MixtilinearIncircles[t],{"(Subscript[O, A])","(Subscript[O, B])","(Subscript[O, C])"},{Red,Blue,Green}}],
{
{Circumcenter[t],"O"}
}
],LabelOffset->.03]
}],AspectRatio->Automatic,PlotRange->All]

Trilinear Vertex Matrix

TrilinearVertexMatrix[MixtilinearTriangle]//TraditionalForm

Derivation

Center/@Circles[MixtilinearIncircles]//InputForm

Area

Factor[Together//@ToSidelengths[Area[MixtilinearTriangle]](Δ/(Area[]))]

ToAngles[%]

Side Lengths

FullSimplify[Refine[Map[Factor,Together//@ToSidelengths[SideLengths[MixtilinearTriangle]],{3}],TriangleQ[]],TriangleQ[]]

Map[Factor,{((a (-a+b+c))/((a+b+c) ))(Sqrt[(((a^3)-5 a (((b-c))^2)+(a^2) (b+c)+3 (((b-c))^2) (b+c))/((a+b-c) (a-b+c)(a+b+c)))]),((b (a-b+c) )/(a+b+c))√((3 (a^3)+(((b-c))^2) (b+3 c)-(a^2) (5 b+3 c)+a ((b^2)+10 b c-3 (c^2)))/((-a+b+c) (a+b-c) (a+b+c))),(((a+b-c) c )/(a+b+c))(Sqrt[((3 (((a-b))^2) (a+b)-5 (((a-b))^2) c+(a+b) (c^2)+(c^3))/((-a+b+c) (a-b+c) (a+b+c)))])},{3}]

{((a (-a+b+c) )/(a+b+c))√(((a^3)+(a^2) b-5 a (b^2)+3 (b^3)+(a^2) c+10 a b c-3 (b^2) c-5 a (c^2)-3 b (c^2)+3 (c^3))/((a+b-c) (a-b+c) (a+b+c))),((b (a-b+c) )/(a+b+c))√((3 (a^3)-5 (a^2) b+a (b^2)+(b^3)-3 (a^2) c+10 a b c+(b^2) c-3 a (c^2)-5 b (c^2)+3 (c^3))/((-a+b+c) (a+b-c) (a+b+c))),(((a+b-c) c )/(a+b+c))√((3 (a^3)-3 (a^2) b-3 a (b^2)+3 (b^3)-5 (a^2) c+10 a b c-5 (b^2) c+a (c^2)+b (c^2)+(c^3))/((-a+b+c) (a-b+c) (a+b+c)))}

Centers

(pts={#,KimberlingLookup[#[MixtilinearTriangle]]}&/@Take[KimberlingCenters,1500]);//Timing

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

l=Map[First,DeleteCases[pts,_?(Last[#]==={}&)],{2}]

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