Yiu Triangle

Author

Eric W. Weisstein
November 8, 2005

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

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

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

Graphic

t=Triangle[Reverse/@{{0,0},{.16,.5},{1,0}}];

Show[Graphics[{
t,
{Red,YiuTriangle[t]},
{Blue,YiuCircles[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{Center/@YiuCircles[t],{"(Subscript[O, A])","(Subscript[O, B])","(Subscript[O, C])"}}]]]
}],AspectRatio->Automatic,PlotRange->All]

Area

area=Factor[Refine[(Together//@ToSidelengths[Area[YiuTriangle]/.Cos[x_]:>TrigExpand[Cos[x]]])/.Sqrt[x_]Sqrt[y_]:>Simplify[Sqrt[x y],TriangleQ[]],TriangleQ[]]]

Factor[Simplify[ToAngles[area](Δ/(Area//ToSidelengths))]]

Side Lengths

abc=Factor/@Simplify[Factor[Refine[(Together//@ToSidelengths[SideLengths[YiuTriangle]/.Cos[x_]:>TrigExpand[Cos[x]]])/.Sqrt[x_]Sqrt[y_]:>Simplify[Sqrt[x y],TriangleQ[]],TriangleQ[]]],TriangleQ[]]

Factor//@abc

Map[Factor,FullSimplify[%6,TriangleQ[]],4]

Map[Factor,Refine[Simplify[%11/. 1/Abs[x_]1/Abs[y_]:>1/ToAngles[Abs[Factor[x y]]],TriangleQ[]],TriangleQ[]],3]

Perspective

ReferenceTriangle, YiuTriangle

Together//@PerspectiveQ[ReferenceTriangle,YiuTriangle]/.Cos[x_]:>TrigExpand[Cos[x]]

Factor[Together//@(ToSidelengths[Perspector[ReferenceTriangle,YiuTriangle]/.Cos[x_]:>TrigExpand[Cos[x]]])/.Sqrt[x_]Sqrt[y_]:>Simplify[Sqrt[x y],TriangleQ[]]]

IncidentCenters[KimberlingCenters,%]//Timing

CenterName[X[5]]

TangentialTriangle, YiuTriangle

Simplify[(Together//@ToSidelengths[PerspectiveQ[TangentialTriangle,YiuTriangle]/.Cos[x_]:>TrigExpand[Cos[x]]])/.Sqrt[x_]Sqrt[y_]:>Simplify[Sqrt[x y],TriangleQ[]]]

Factor[(Together//@ToSidelengths[Perspector[TangentialTriangle,YiuTriangle]/.Cos[x_]:>TrigExpand[Cos[x]]])/.Sqrt[x_]Sqrt[y_]:>Simplify[Sqrt[x y],TriangleQ[]]]

IncidentCenters[KimberlingCenters,%2]//Timing

CenterName[X[195]]

Derivation

Center/@Circles[YiuCircles]

Centers

(pts={#,KimberlingLookup[#[YiuTriangle]]}&/@Take[KimberlingCenters,1000]);//Timing

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

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

(pts={#,KimberlingLookup[#[YiuTriangle]]}&/@Take[KimberlingCenters,{1001,1500}]);//Timing

...crashed...

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

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