Miquel Circles

Author

Eric W. Weisstein
December 14, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.2,.7}}];
x={.7,.6,.5};
Show[Graphics[{
t,
{Red,#[t]&/@MiquelCircles[x]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{Cartesian[#[t]]&/@SideRatioPoints[x],{"(Subscript[P, A])","(Subscript[P, B])","(Subscript[P, C])"}}],
{{Cartesian[MiquelPoint[x][t]],"M",Blue}}
],LabelOffset->.04,PointSize[.025]]
}],AspectRatio->Automatic,PlotRange->All]

Derivation

Simplify[Distance[{0,0,1},{alpha,0,gamma}],TriangleQ[]&&Area>0&&beta>0&&alpha>0&&gamma>0]

{alpha,beta,gamma}/.Simplify[Solve[{kb b==((a alpha b)/(a alpha+c gamma))&&2Area==(a alpha+b beta+c gamma)&&beta==0},{alpha,beta,gamma}]]

SideRatioPoints[{ka_,kb_,kc_}]:={{0,(1-ka)c,ka b},{kb c,0,(1-kb)a},{(1-kc)b,kc a,0}}

SideRatioPoints[{.76,.4,.4}]

circs=Simplify[Circumcircle/@Triangle/@{
{{1,0,0},{(1-kc)b,kc a,0},{kb c,0,(1-kb)a}},
{{0,1,0},{(1-kc)b,kc a,0},{0,(1-ka)c,ka b}},
{{0,0,1},{0,(1-ka)c,ka b},{kb c,0,(1-kb)a}}
},Area>0&&0<ka<1&&0<kb<1&&0<kc<1&&TriangleQ[]]

circs2=circs/.Circle[ctr_List,r_]:>Circle[Symmetrize[ctr],Simplify[r/.Sqrt[s_]:>Sqrt[ToAngles[Factor[s]]],Area>0]]

MiquelCircles[{ka_,kb_,kc_}]:={Circle[{(a^4)-(a^2) (b^2)-2 (a^2) (c^2)-(b^2) (c^2)+(c^4)-(a^2) (b^2) kb+(b^4) kb-(b^2) (c^2) kb+(a^2) (c^2) kc+(b^2) (c^2) kc-(c^4) kc,a b (-(a^2)+(b^2)+(c^2)+(a^2) kb-(b^2) kb-(c^2) kb-2 (c^2) kc),-a c (2 (b^2)-2 (b^2) kb+(a^2) kc-(b^2) kc-(c^2) kc)},((b c (Sqrt[(b^2) (-1+kb) (-1+kb+kc)+kc (-(a^2) (-1+kb)+(c^2) (-1+kb+kc))]))/(4 Area))],Circle[{((2 (c^2)-(a^2) ka+(b^2) ka-(c^2) ka-2 (c^2) kc)/c),((-(a^4)+2 (a^2) (b^2)-(b^4)+(a^2) (c^2)+(b^2) (c^2)+(a^4) ka-(a^2) (b^2) ka-(a^2) (c^2) ka+(a^2) (c^2) kc+(b^2) (c^2) kc-(c^4) kc)/(a b c)),((-(a^2)+(b^2)-(c^2)+2 (a^2) ka+(a^2) kc-(b^2) kc+(c^2) kc)/a)},((a c (Sqrt[(c^2) (-1+kc) (-1+ka+kc)+ka (-(b^2) (-1+kc)+(a^2) (-1+ka+kc))]))/(4 Area))],Circle[{((-(a^2)-(b^2)+(c^2)+(a^2) ka+(b^2) ka-(c^2) ka+2 (b^2) kb)/b),((2 (a^2)-2 (a^2) ka-(a^2) kb-(b^2) kb+(c^2) kb)/a),(((a^2) (b^2)-(b^4)+(a^2) (c^2)+2 (b^2) (c^2)-(c^4)-(a^4) ka+(a^2) (b^2) ka+(a^2) (c^2) ka-(a^2) (b^2) kb+(b^4) kb-(b^2) (c^2) kb)/(a b c))},((a b (Sqrt[(a^2) (-1+ka) (-1+ka+kb)+kb (-(c^2) (-1+ka)+(b^2) (-1+ka+kb))]))/(4 Area))]}

#[t]&/@MiquelCircles[{.76,.4,.4}]

Intersections

?Intersections

Intersections@@Take[MiquelCircles[{ka,kb,kc}],2]

Factor[Together//@ToSidelengths[RadicalCenter[MiquelCircles[{ka,kb,kc}]]]]

Symmetrize[%]

{a,b,c}(Collect[#,{a^2,b^2,c^2},Factor]&/@(%302/{a,b,c}))//ColumnForm

InputForm[%]

MiquelPoint[{ka_,kb_,kc_}]:={a*(a^2*(-1+ka)*ka+b^2*ka*kb+c^2*(-1+ka)*(-1+kc)),b*(a^2*(-1+ka)*(-1+kb)+b^2*(-1+kb)*kb+c^2*kb*kc),c*(b^2*(-1+kb)*(-1+kc)+a^2*ka*kc+c^2*(-1+kc)*kc)}