Malfatti's Problem

Author

Eric W. Weisstein
November 27, 2004

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

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

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

Related Centers

am=Select[KimberlingCenters,StringMatchQ[CenterName[#],"*Malfatti*"]&]

CenterName/@am

CenterFunction/@am

Figure

Unit[v_]:=v/Norm[v]

MalfattiCircles[t_Triangle]:=Module[{eqns,soln,r1,r2,r3,d1,d2,d3,a,b,c
},
{a,b,c}=SideLengths[Evaluate[t]];
eqns={
4r1 r2==(d1+d2-c)^2,
4r2 r3==(d2+d3-a)^2,
4r1 r3==(d1+d3-b)^2,
2 b c ((d1^2)-(r1^2))==((d1^2)+(r1^2))(-a^2+(b^2) +(c^2) ),2 a c ((d2^2)-(r2^2))== ((d2^2)+(r2^2))(a^2-(b^2)+(c^2)),2 a b ((d3^2)-(r3^2))== ((d3^2)+(r3^2))(a^2+b^2-c^2)
};
Off[FindRoot::"precw"];
soln=Last/@FindRoot[eqns,Evaluate[Sequence@@({#,Max[a,b,c]/3}&/@{r1,r2,r3,d1,d2,d3})],WorkingPrecision->20];
On[FindRoot::"precw"];
Circle[#1-Sqrt[#2^2+#3^2]#4,#2]&@@@Transpose[{Vertices[t],Sequence@@Partition[soln,3],Unit/@Subtract@@@(Coordinates/@AngleBisectors[t])}]
]

t=Triangle[{{0,0},{1,0},{.1,.6}}];
circ=MalfattiCircles[t];
m=CircleCenter/@circ;
Show[Graphics[{
{Blue,circ=MalfattiCircles[t]},
t,
{Red,
PointLabels[Transpose[{m,{(Subscript[Γ, A]),(Subscript[Γ, B]),(Subscript[Γ, C])}}],
LabelOffset->.03]
},
PointLabels[Transpose[{VertexPoints[t],{"A","B","C"}}],
LabelOffset->.03]
}],AspectRatio->Automatic,PlotRange->All]