Ajima-Malfatti Points

Author

Eric W. Weisstein
November 27, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/Ajima-MalfattiPoints.nb.

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

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

Related Centers

am=Select[X/@Range[2667],StringMatchQ[CenterName[#],"*Malfatti*"]&]

CenterName/@am

CenterFunction/@am

First Ajima-Malfatti Point

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])}]
]

CenterFunction[X[179]]

CircleIntersections[t_Triangle]:=Module[{circs=MalfattiCircles[t]},
Point[Coordinates[CircleCenter[#[[1]]]]-CircleRadius[#[[1]]]Unit[Subtract@@(Coordinates/@CircleCenter/@#)]]&/@Partition[RotateLeft[circs],2,1,1]
]

t=Triangle[{{0,0},{1,0},{.1,.6}}];
circ=MalfattiCircles[t];
m=CircleCenter/@circ;
Show[Graphics[{
{Blue,circ=MalfattiCircles[t]},
t,
{Red,Cevians[t,X[179][t]]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{m,{(Subscript[Γ, A]),(Subscript[Γ, B]),(Subscript[Γ, C])}}],
Append[#,Red]&/@Transpose[{CircleIntersections[t],{"A'","B'","C'"}}],
{{X[179][t],"(Subscript[X, 179])",Red}}
],
LabelOffset->.03]
}],AspectRatio->Automatic,PlotRange->All]

Second Ajima-Malfatti Point

Center

f(A,B,C) = 1/t(B,C,A) + 1/t(C,B,A) - 1/t(C,A,B),
t(A,B,C) = 1 + 2(sec A/4 cos B/4 cos C/4)2

t[A_,B_,C_]:=1+2(Sec[A/4]Cos[B/4]Cos[C/4])^2

(1/(t[B,C,A]))+(1/(t[C,B,A]))-(1/(t[C,A,B]))

(1/(t[C,B,A]))==(1/(t[C,A,B]))

CenterFunction[X[180]]

Intersections@@@Partition[Line/@Map[Coordinates,Transpose[{CircleIntersections[t],Excenters[t]}],{2}],2,1,1]

i=%148[[1]]

X[180][t]

ETC is wrong

Figure

Show[Graphics[{
{Green,Excircles[t],{Dashing[{.01}],Sides[t,1]}},
t,
{Blue,circ=MalfattiCircles[t]},
{Red,Line[Coordinates/@{i,#}]&/@Excenters[t]},
PointLabels[Join[
Append[#,Red]&/@Transpose[{CircleIntersections[t],{"A'","B'","C'"}}],
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Green]&/@Transpose[{Excenters[t],{"A''","B''","C''"}}],
{{i,"(Subscript[X, 180])",{LabelOffset->-.035,Red}}}
],
LabelOffset->.03,PointSize[.02]]
}],AspectRatio->Automatic,
PlotRange->{{-.1,1.1},{-.1,.7}},ImageSize->400]