Kenmotu Circle

Author

Eric W. Weisstein
November 17, 2005

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

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

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

Graphic

t=Triangle[{{0,0},{1,0},{.4,.7}}];
Show[Graphics[{
t,
{Blue,KenmotuSquares[t]},
KenmotuCircle[t],
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{
{KenmotuPoint[t],"Ke",Red}
}
],LabelOffset->.04]
}],AspectRatio->Automatic,PlotRange->All]

As A Tucker Circle

circ=TuckerCircle[Pi/4]

Center

KimberlingLookup[Center[circ]]

Radius

Factor[TrigExpand[Circumradius Csc[BrocardAngle+(π/4)] Sin[BrocardAngle]]]

Refine[ToAngles[FullSimplify[ToSidelengths[%80],TriangleQ[]]],Area>0]

Circle Function

Together//@Expand[Together//@ToSidelengths[TrigExpand[cf=CircleFunction[circ][[1]]]]/.Sqrt[s_]Sqrt[t_]:>Simplify[Sqrt[s t],TriangleQ[]]/.Sqrt[s_]:>Simplify[Sqrt[s],TriangleQ[]]]/.Sqrt[s_]:>Sqrt[Factor[s]]

{numer,denom}=Expand[Through[{Numerator,Denominator}[%161]]];

coef=(Sqrt[-((a-b-c) (a+b-c) (a-b+c) (a+b+c))]);

d1=Factor[Coefficient[denom,coef]]

d2=Factor[denom-d1 coef]

d2

d3=PolynomialGCD[d1,d2]

d3=4 (a-b-c) (a+b-c) (a-b+c) (a+b+c)

n1=Factor[Coefficient[numer,coef]]

n2=Factor[numer-n1 coef]

n3=PolynomialGCD[n1,n2]

n3=4 b (a-b-c) (a+b-c) c (a-b+c) (a+b+c)

n3/d3

((Cancel[((n1 coef)/n3)]+Cancel[(n2/n3)])/(Cancel[((d1 coef)/d3)]+Cancel[(d2/d3)]))b c

l=b c(-(a^6)-(a^4) (b^2)+3 (a^2) (b^4)-(b^6)-(a^4) (c^2)+8 (a^2) (b^2) (c^2)+3 (b^4) (c^2)+3 (a^2) (c^4)+3 (b^2) (c^4)-(c^6)-(Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) ((a^4)-2 (a^2) (b^2)-(b^4)-2 (a^2) (c^2)-4 (b^2) (c^2)-(c^4)))/((a^8)-6 (a^4) (b^4)+(b^8)-16 (a^4) (b^2) (c^2)-16 (a^2) (b^4) (c^2)-6 (a^4) (c^4)-16 (a^2) (b^2) (c^4)-6 (b^4) (c^4)+(c^8)-4 (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) ((a^2)+(b^2)+(c^2)) ((a^2) (b^2)+(a^2) (c^2)+(b^2) (c^2)))

ll=Factor[Expand[(b c (-(a^6)-(a^4) (b^2)+3 (a^2) (b^4)-(b^6)-(a^4) (c^2)+8 (a^2) (b^2) (c^2)+3 (b^4) (c^2)+3 (a^2) (c^4)+3 (b^2) (c^4)-(c^6)-(Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) ((a^4)-2 (a^2) (b^2)-(b^4)-2 (a^2) (c^2)-4 (b^2) (c^2)-(c^4))))((a^8)-6 (a^4) (b^4)+(b^8)-16 (a^4) (b^2) (c^2)-16 (a^2) (b^4) (c^2)-6 (a^4) (c^4)-16 (a^2) (b^2) (c^4)-6 (b^4) (c^4)+(c^8)+4 (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) ((a^2)+(b^2)+(c^2)) ((a^2) (b^2)+(a^2) (c^2)+(b^2) (c^2)))]/Expand[(((a^8)-6 (a^4) (b^4)+(b^8)-16 (a^4) (b^2) (c^2)-16 (a^2) (b^4) (c^2)-6 (a^4) (c^4)-16 (a^2) (b^2) (c^4)-6 (b^4) (c^4)+(c^8)-4 (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) ((a^2)+(b^2)+(c^2)) ((a^2) (b^2)+(a^2) (c^2)+(b^2) (c^2)))((a^8)-6 (a^4) (b^4)+(b^8)-16 (a^4) (b^2) (c^2)-16 (a^2) (b^4) (c^2)-6 (a^4) (c^4)-16 (a^2) (b^2) (c^4)-6 (b^4) (c^4)+(c^8)+4 (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) ((a^2)+(b^2)+(c^2)) ((a^2) (b^2)+(a^2) (c^2)+(b^2) (c^2))))]]

lll=-((b c)/((((a^4)+(b^4)+(c^4)))^2))(Factor[4Area Coefficient[(((a^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+2 (a^2) (b^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-(b^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+2 (a^2) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-(c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]))),(Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])]]+Factor[ ((a^6)-3 (a^4) (b^2)+(a^2) (b^4)+(b^6)-3 (a^4) (c^2)-4 (a^2) (b^2) (c^2)+(b^4) (c^2)+(a^2) (c^4)+(b^2) (c^4)+(c^6))])

t=RandomTriangle[20];

ToSidelengths[cf]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

ToSidelengths[l]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

ToSidelengths[lll]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

InputForm[lll]

KimberlingLookup[CyclicTrilinears[CircleFunction[KenmotuCircle]]]

Name[X[492]]

CenterFunction[X[492]]

CircleFunction[KenmotuCircle]

FullSimplify[ToSidelengths[((CenterFunction[X[492]])/(CircleFunction[KenmotuCircle]))],TriangleQ[]]

Factor[-a^6+(a^4) ((b^2)+(c^2))-((((b^2)-(c^2)))^2) ((b^2)+(c^2))+(a^2) ((b^4)+6 (b^2) (c^2)+(c^4))]

Factor[2 (a^2) (b^2)+2 (a^2) (c^2)+2 (b^2) (c^2)]

-(Area/( (a^2) (b^2) (c^2)))(Area ((a^2)+(b^2)+(c^2))+2 ((a^2) (b^2)+(a^2) (c^2)+(b^2) (c^2)))

Incident Centers

(pts=IncidentCenters[Take[KimberlingCenters,All],KenmotuCircle])//Timing