In[]:=
ConwayCircle::usage="From a base triangle, the incircle and incenter are obtained, then each side of the triangle extends the same length as the original side to a point. All these points are concyclic and form the Conway circumference. In the command below, enter the coordinates of the points in clockwise direction. References: Grinberg, Darij and Weisstein, Eric W. Conway Circle. From ​​ MathWorld; Personal.";
In[]:=
ConwayCircle[tri={{x1_,y1_},{x2_,y2_},{x3_,y3_}}]:=​​Module{},Test=Det[{{x1,y1,1},{x2,y2,1},{x3,y3,1}}];​​IfTest==0,Print[PasteButton["It's not a triangle"]],​​A={x1,y1};B={x2,y2};C1={x3,y3};​​c11=Norm[A-B];b11=Norm[B-C1];a1=Norm[A-C1];​​r1=
c11
c11+a1
;xAc=x1+
x2-x1
r1
;yAc=y1+
y2-y1
r1
;r2=
c11
c11+b11
;xBc=x2+
x1-x2
r2
;yBc=y2+
y1-y2
r2
;​​r3=
a1
b11+a1
;xBa=x3+
x1-x3
r3
;yBa=y3+
y1-y3
r3
;r4=
a1
a1+c11
;xCa=x1+
x3-x1
r4
;yCa=y1+
y3-y1
r4
;​​r5=
a1
a1+b11
;​​xAb=
r5x3-x2
r5-1
;yAb=
r5y3-y2
r5-1
;r6=
b11
b11+c11
;xCb=x2+
x3-x2
r6
;yCb=y2+
y3-y2
r6
;​​Ac={xAc,yAc};Bc={xBc,yBc};Ba={xBa,yBa};Ca={xCa,yCa};Ab={xAb,yAb};Cb={xCb,yCb};​​γ=TriangleCenter[tri,"Incenter"];​​xin=γ[[1]];yin=γ[[2]];​​I1={xin,yin};​​γ1=TriangleMeasurement[tri,"Inradius"];​​rin=γ1;​​rcirc=Norm[I1-Ab];​​Print[" "];​​Print[PasteButton["The coordinates of the concyclic points are:"]];​​Print[" "];​​l=Grid[{{
PointA
c
,
PointB
c
,
PointA
a
,
PointC
a
,
PointB
b
,
PointC
b
},{N[Bc],N[Ac],N[Ba],N[Ca],N[Ab],N[Cb]}},ItemSize15,ItemStyle15,FrameAll];​​Print[l];​​Print[" "];​​Print[PasteButton["The center and radius of the Conway's circle (red colored):"]];​​Print[" "];​​m=Grid[{{Center,Radius},{N[I1],N[rcirc]}},ItemSize15,ItemStyle15,FrameAll];​​Print[m];​​Print[" "];​​label[sym_,Point[pos_]]:={PointSize[Scaled[0.018]],Hue[1,1,1],Point[pos],Text[Style[sym,White,FontSizeScaled[0.012]],pos]};​​Show[{Graphics[{LightYellow,EdgeForm[{Thick,Black}],Polygon[{A,B,C1}]}],Graphics[{Black,Thick,Circle[I1,rin]}],Graphics[{Red,Thick,Circle[I1,rcirc]}],Graphics[{Black,Thick,Dashed,Line[{B,Ab}]}],Graphics[{Black,Thick,Dashed,Line[{B,Ac}]}],Graphics[{Black,Thick,Dashed,Line[{A,Ba}]}],Graphics[{Black,Thick,Dashed,Line[{A,Bc}]}],Graphics[{Black,Thick,Dashed,Line[{C1,Ca}]}],Graphics[{Black,Thick,Dashed,Line[{C1,Cb}]}],Graphics[{MapThread[label,{{"A","B","C","
B
c
","
A
c
","
A
a
","
C
a
","
B
b
","
C
b
","In"},{Point[A],Point[B],Point[C1],Point[Ac],Point[Bc],Point[Ba],Point[Ca],Point[Ab],Point[Cb],Point[I1]}}]}]},AxesTrue,BoxRatiosAutomatic,AxesOrigin{0,0},ImageSize900,PlotRangeAll]
In[]:=
ConwayCircle[tri={{-2,5},{7,12},{9,-4}}]
The coordinates of the concyclic points are:
PointA
c
PointB
c
PointA
a
PointC
a
PointB
b
PointC
b
{-14.7279,-4.89949}
{18.2188,20.7257}
{-14.4797,15.2107}
{17.8245,-11.22}
{5.23714,26.1029}
{10.4142,-15.3137}
The center and radius of the Conway's circle (red colored):
Center
Radius
{4.06947,4.92508}
21.21
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