In[]:=
FeuerbachHyperbola::usage="Feuerbach's point is the point where Euler's circle (or 9 points) is tangent to the circle inscribed in the triangle. The Feuerbach hyperbola has Feuerbach's point as its center, passes through the three vertices of the triangle, passes through the incenter, the orthocenter, the Nagel point, the Gergonne point, the mittenpunkt and many other important points. References: Wolfram Mathworld: Feuerbach hyperbola; Personal. ";FeuerbachHyperbola[tri={{x11_,y11_},{x2_,y2_},{x3_,y3_}},{X0_,X1_},{Y0_,Y1_}]:=Module{},DEP=Det[{{x11,y11,1},{x2,y2,1},{x3,y3,1}}];IfDEP==0||X0≥X1||Y0≥Y1,Print[PasteButton["Incorrect data or points do not form a triangle"]],xm1=;ym1=;xm2=;ym2=;xm3=;ym3=;ρ=TriangleCenter[tri,"Orthocenter"];xo=ρ[[1]];yo=ρ[[2]];tri1={{xm1,ym1},{xm2,ym2},{xm3,ym3}};β=TriangleCenter[tri1,"Circumcenter"];xce=β[[1]];yce=β[[2]];β1=TriangleMeasurement[tri1,"Circumradius"];rce=β1;γ=TriangleCenter[tri,"Incenter"];xin=γ[[1]];yin=γ[[2]];γ1=TriangleMeasurement[tri,"Inradius"];rin=γ1;s4=Solve[{+==,+==},{x,y}];xff=x/.Take[s4,{1}];xf=xff.{1};yff=y/.Take[s4,{1}];yf=yff.{1};Print[""];Print[PasteButton["Coordinates of the Feuerbach point"]];Print["(",N[xf]," , ",N[yf],")"];Print[""];s5=NSolve[aex11^2+x11y11+cey11^2+dex11+eey11+fe==0, aex2^2+x2y2+cey2^2+dex2+eey2+fe==0, aex3^2+x3y3+cey3^2+dex3+eey3+fe==0, aexin^2+xinyin+ceyin^2+dexin+eeyin+fe==0, aexo^2+xoyo+ceyo^2+dexo+eeyo+fe==0,{ae,ce,de,ee,fe}]; a11=ae/.s5;c11=ce/.s5;d11=de/.s5;e11=ee/.s5;f11=fe/.s5; at=Dot[a11,{1}]; bt=1; ct=Dot[c11,{1}]; dt=Dot[d11,{1}]; et=Dot[e11,{1}]; ft2=Dot[f11,{1}]; Print[PasteButton["The equation of Feuerbach's hyperbola is:"]];Print[" "];Print[{at,bt,ct,dt,et,ft2}.{,xy,,x,y,1},"= 0"];Print[""];Print[PasteButton["Information about Feuerbach's hyperbola:"]];p5={{x11,y11},{x2,y2},{x3,y3},{xin,yin},{xo,yo}};δ=ResourceFunction["ConicProperties"][{at,bt,ct,dt,et,ft2}.{,xy,,x,y,1}==0,{x,y}];Print[δ];Print[" "];Conicas[at,bt,ct,dt,et,ft2,{X0,X1},{Y0,Y1}];g1=Line[{{x11,y11},{x2,y2},{x3,y3},{x11,y11}}];g2=Point[{xo,yo}];g3=Point[{xin,yin}];g4=Point[{x11,y11}];g5=Point[{x2,y2}];g6=Point[{x3,y3}];g7=Point[{xf,yf}];g8=ContourPlot[atx^2+btxy+cty^2+dtx+ety+ft2==0,{x,X0,X1},{y,Y0,Y1}];g9=Circle[{xce,yce},rce];g10=Circle[{xin,yin},rin];SwatchLegend[{Red,Green,Orange,Gray,Purple},{"Feuerbach's point","Orthocenter","Incenter","Nine points circle","Inscribed circumference"}]Show[{Graphics[{Green,PointSize[0.01],g2}],Graphics[{Black,g1}],Graphics[{Orange,PointSize[0.01],g3}],Graphics[{Black,PointSize[0.01],g4}],Graphics[{Black,PointSize[0.01],g5}],Graphics[{Black,PointSize[0.01],g6}],Graphics[{Red,PointSize[0.01],g7}],g8,Graphics[{Gray,g9}],Graphics[{Purple,g10}]},AxesTrue,AxesOrigin{0,0},ImageSize900,PlotLabelStyle[Framed["Feuerbach's hyperbola and details of triangle"],16,Blue,BackgroundYellow]]
x11+x2
2
y11+y2
2
x2+x3
2
y2+y3
2
x3+x11
2
y3+y11
2
2
(x-xce)
2
(y-yce)
2
rce
2
(x-xin)
2
(y-yin)
2
rin
2
x
2
y
2
x
2
y
In the command below, type the new coordinates of the points. You can also change the extension to X and Y. Evaluate the notebook.
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FeuerbachHyperbola[tri={{-2,5},{7,12},{9,-4}},{-20,20},{-20,20}]
(5.78404 , 8.29995)
51.4552-4.34115x-0.342217-11.4648y+xy+0.342217= 0
2
x
2
y
TypeHyperbola,Equation51.4552-4.34115x-0.342217-11.4648y+xy+0.3422170,Foci{{3.28741,3.56576},{8.28067,13.0341}},Vertices{{4.01866,4.95237},{7.54943,11.6475}},Eccentricity1.41421,Asymptotes{y26.9915-3.23157x,y6.51009+0.309447x},Center{5.78404,8.29995},SemimajorAxisLength3.78455,SemiminorAxisLength3.78455,Directrices{y14.3175-0.527362x,y14.3635-0.527362x},AxesOfSymmetry{y-2.66793+1.89623x,y11.3502-0.527362x},SemiLatusRectum3.78455,RotationAngle1.0855,ConjugateHyperbola8.99555×x+7.09129×+2.37569×y-2.07216×xy-7.09129×4.05608×,BaseConicEquation0.0698185-0.06981851,RotationAngle1.0855,DisplacementFromOrigin{5.78404,8.29995}
2
x
2
y
38
10
37
10
2
x
39
10
38
10
37
10
2
y
40
10
2
x
2
y
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