Thébault'sTheoremIII::usage="This animation shows that Thébault's theorem III is felfilled for any cevian. Theorem III states: It states that given any triangle ABC with a point M on the BC side, if two tangent circles to AM, BC and circumscribed circle of the triangle are constructed, its centers and the center of the inscribed circle of ABC are collinear. ";
In[]:=
Manipulatex1=-2;y1=5;x2=7;y2=12;x3=9;y3=-4;xc=
488
79
;yc=
298
79
;rc=
65
101
79
;​​xi=
-
65
-
130
9
-
37
130
99
-
59
202
99
+-
59
65
9
+
68
130
9
130
9
+
202
11
65
-
130
9
130
11
-
7
202
99
--
7
65
9
-
8
130
9
130
9
+
202
11
;​​yi=
-
7
65
9
-
8
130
9
-
37
130
99
-
59
202
99
--
59
65
9
+
68
130
9
130
11
-
7
202
99
65
-
130
9
130
11
-
7
202
99
--
7
65
9
-
8
130
9
130
9
+
202
11
;​​ri=
1
202
11-
37
11
-
--
7
65
9
-
8
130
9
-
37
130
99
-
59
202
99
-
59
65
9
-
68
130
9
130
11
-
7
202
99
65
-
130
9
130
11
-
7
202
99
--
7
65
9
-
8
130
9
130
9
+
202
11
+
9-
65
+
130
9
-
37
130
99
-
59
202
99
+-
59
65
9
+
68
130
9
130
9
+
202
11
11
65
-
130
9
130
11
-
7
202
99
--
7
65
9
-
8
130
9
130
9
+
202
11
;​​xr=
x1+rx3
1+r
;yr=
y1+ry3
1+r
;​​A1=y2-yr;B1=xr-x2;C1=x2yr-y2xr;​​A3=y1-y3;B3=x3-x1;C3=x1y3-x3y1;​​​​s1=NSolve
A3h+B3k+C3
2
A3
+
2
B3
==
A1h+B1k+C1
2
A1
+
2
B1
,
2
(h-xc)
+
2
(k-yc)
+
A1h+B1k+C1
2
A1
+
2
B1
==rc,{h,k};​​If[r<0.518,t=1,t=2];​​h11=h/.Take[s1,{t}];h1=h11.{1};k11=k/.Take[s1,{t}];k1=k11.{1};r1=
Abs[A1h1+B1k1+C1]
2
A1
+
2
B1
;​​s2=NSolve
A3h+B3k+C3
2
A3
+
2
B3
==-
A1h+B1k+C1
2
A1
+
2
B1
,
2
(h-xc)
+
2
(k-yc)
-
A1h+B1k+C1
2
A1
+
2
B1
==rc,{h,k};​​If[r<0.518,p=2,p=1];​​h22=h/.Take[s2,{p}];h2=h22.{1};k22=k/.Take[s2,{p}];k2=k22.{1};r2=
Abs[A1h2+B1k2+C1]
2
A1
+
2
B1
;​​g1=Line[{{x1,y1},{x2,y2},{x3,y3},{x1,y1}}];​​g2=Point[{xi,yi}];​​g3=Point[{h1,k1}];​​g4=Point[{h2,k2}];​​g7=Circle[{xc,yc},rc];g8=Circle[{h1,k1},r1];g9=Circle[{h2,k2},r2];​​g10=Line[{{xr,yr},{x2,y2}}];​​g11=InfiniteLine[{{h1,k1},{h2,k2}}];​​g12=Circle[{xi,yi},ri];​​g13=Point[{xr,yr}];​​w=Grid[{{Center,Radius},{{h1,k1},N[r1]}},FrameAll,FrameStyle->Purple,ItemSize->15,ItemStyle->12];​​o=Grid[{{Center,Radius},{{h2,k2},N[r2]}},FrameAll,FrameStyle->Green,ItemSize->15,ItemStyle->12];​​Show[{Graphics[{Blue,g1}],Graphics[{Black,Thick,PointSize[0.011],g2}],Graphics[{Purple,Thick,PointSize[0.011],g3}],Graphics[{Green,Thick,PointSize[0.011],g4}],Graphics[{Brown,g7}],Graphics[{Purple,g8}],Graphics[{Green,g9}],Graphics[{Black,g10}],Graphics[{Black,Dashed,g11}],Graphics[{Black,Dashed,g12}],Graphics[{Red,Thick,PointSize[0.011],g13}]},AxesTrue,AxesOrigin{0,0},BoxRatiosAutomatic,PlotRangeAll,PerformanceGoal"Quality",ImageSize900,PlotRange{{-2,13},{-2,12}}],Style["Thébault's Theorem III:",Bold,Medium],{{r,0,"Ratio (r)"},0,10,0.2},Delimiter,{{o,2,"Circumference 1"}},Delimiter,{{w,3,"Circumference 2"}},ControlPlacementTop
Out[]=
​
Thébault's Theorem III:
Ratio (r)
Circumference 1
Center
Radius
{1.78722,6.1404}
3.28084
Circumference 2
Center
Radius
{9.15473,2.21712}
4.90977