Thales’s theorem and tangent to a circle
MathWord: An inscribed angle in a semicircle is a right angle.
In this demonstration the Thales’s theorem is used to construct a tangent to a circle.
In this demonstration the Thales’s theorem is used to construct a tangent to a circle.
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Manipulate[(*adistheradiusofthesecondcircleinthefunctionoftheanglealf*)ad=1/(2Cos[Pi-alf]);spad={Red,Sphere[{ad,0,0},0.07]};gam=ArcSin[1/(2ad)];h=(1+2ad)Tan[gam];v0={0,0,0};vA={-1,0,0};vB={1,0,0};vC={-Cos[alf],Sin[alf],0};vD={Cos[alf],-Sin[alf],0};vE={2ad,0,0};vF={-1,h,0};(*ciristheoriginalcircle*)cir=ParametricPlot3D[{Sin[x],Cos[x],0},{x,0,2Pi},PlotStyle->{Yellow,Tube[0.02]}][[1]];spA=Sphere[vA,0.05];spB=Sphere[vB,0.05];spC=Sphere[vC,0.05];spD=Sphere[vD,0.05];spE=Sphere[vE,0.05];dia=Cylinder[{vA,vB},0.02];dia1=Cylinder[{vC,vD},0.02];diaad=Cylinder[{vB,vE},0.02];(*tanisthetangent*)tan=Cylinder[{vF,vE},0.03];(*centistheradiustowhichthetangentiscosntructed*)cent=Cylinder[{vC,v0},0.04];(*pol1andpol3arethetwotriangleswithrightangles*)pol1={Yellow,Tube[{vA,vB,vC,vA}]};pol3=Polygon[{vE,vC,v0}];(*pol2isthereversetriangle*);pol2=Polygon[{vA,vB,vD}];(*circleisthesecondcircle*)circle=ParametricPlot3D[{adSin[x]+ad,adCos[x],0},{x,0,2Pi},PlotStyle->{Cyan,Tube[0.02]}][[1]];Graphics3D[{spad,dia,{Red,tan},circle,diaad,pol1,Purple,spA,spB,spC,spE,cir,LightGreen,pol3,Red,cent},Boxed->False,SphericalRegion->True,ViewPoint->{0,0,100},PlotRange->{{-1.3,5},{-3,3},{-0.2,0.2}},ViewAngle->0.01],{{alf,1.8,"angle"},1,2.5,Appearance->"Labeled"},TrackedSymbols:>alf]
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Graphics3D[{cent,pol1,pol2,pol3}]
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ad=1/(2Sin[0.43])
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1.19941