AngleBetweenCircles::usage="Show that the circles circumscribed to two faces of a tetrahedron intersect at the same angle as the circles circumscribed to the other two. References: Usunáriz Balanzategui, Ubaldo; Usunáriz Sala, Ignacio. Problemas de Geometría​; Personal.";
Development:
Cartesian equation of the
P
2
P
3
P
4
plane:
In[]:=
P4={-1,-1,1};P3={-1,1,-1};P2={1,-1,-1};​​C1=
P2+P3+P4
3
​​r1=Norm[C1-P2]​​
-
1
3
,-
1
3
,-
1
3
;
2
2
3
;
{-1,-1,1}+u{0,2,-2}+v{2,0,-2};​​Eliminate[{x==-1+2v,y==-1+2u,z==1-2u-2v},{u,v}]
In[]:=
-1-y-zx;​​y=u;​​z=v;​​{-1,1,0}{1,1,1}
{1,1,-2};
ζ
1
[t_]:=-
1
3
,-
1
3
,-
1
3
+
2
2
3
Cos[t]
Norm[{-1,1,0}]
{-1,1,0}+
2
2
3
Sin[t]
Norm[{1,1,-2}]
{1,1,-2};
Cartesian equation of the
P
1
P
2
P
4
plane:
In[]:=
P4={-1,-1,1};P2={1,-1,-1};P1={1,1,1};​​C2=
P1+P2+P4
3
​​r2=Norm[C2-P1]

1
3
,-
1
3
,
1
3
;
2
2
3
;
In[]:=
{-1,-1,1}+u{2,0,-2}+v{2,2,0};​​Eliminate[{x==-1+2u+2v,y==-1+2v,z==1-2u},{u,v}]​​1+y-zx;​​y=u;​​z=v;​​{1,1,0}{1,-1,1}
{1,-1,-2};
ζ
2
[t_]:=
1
3
,-
1
3
,
1
3
+
2
2
3
Cos[t]
Norm[{1,1,0}]
{1,1,0}+
2
2
3
Sin[t]
Norm[{1,-1,-2}]
{1,-1,-2};
Cartesian equation of the
P
1
P
3
P
4
plane:
In[]:=
P4={-1,-1,1};P1={1,1,1};P3={-1,1,-1};​​C3=
P1+P3+P4
3
​​r3=Norm[C3-P3]
-
1
3
,
1
3
,
1
3
;
2
2
3
;
In[]:=
{-1,-1,1}+u{2,2,0}+v{0,2,-2};​​Eliminate[{x==-1+2u,y==-1+2u+2v,z==1-2v},{u,v}]​​-1+y+zx;​​y=u;​​z=v;​​{1,1,0}{1,-1,-1}​​
{-1,1,-2};
ζ
3
[t_]:=-
1
3
,
1
3
,
1
3
+
2
2
3
Cos[t]
Norm[{1,1,0}]
{1,1,0}+
2
2
3
Sin[t]
Norm[{-1,1,-2}]
{-1,1,-2};
Cartesian equation of the
P
1
P
2
P
3
plane:
In[]:=
P2={1,-1,-1};P1={1,1,1};P3={-1,1,-1};​​C4=
P1+P2+P3
3
​​r4=Norm[C4-P2]​​

1
3
,
1
3
,-
1
3
;
2
2
3
;
In[]:=
{1,-1,-1}+u{0,2,2}+v{-2,2,0};​​Eliminate[{x==1-2v,y==-1+2u+2v,z==-1+2u},{u,v}]​​1-y+zx;​​y=u;​​z=v;​​{-1,1,0}{1,1,-1}
{-1,-1,-2};
ζ
4
[t_]:=
1
3
,
1
3
,-
1
3
+
2
2
3
Cos[t]
Norm[{-1,1,0}]
{-1,1,0}+
2
2
3
Sin[t]
Norm[{-1,-1,-2}]
{-1,-1,-2};
Angle Orange-Purple:
In[]:=
α
1
=N
180
π
ArcCos
{1,-1,-1}.{1,1,1}
Norm[{1,-1,-1}]Norm[{1,1,1}]

Out[]=
109.471
But:
In[]:=
180-109.47122063449069`
Out[]=
70.5288
Angle DarkGreen-Brown:
In[]:=
α
1
=N
180
π
ArcCos
{1,-1,1}.{1,1,-1}
Norm[{1,-1,1}]Norm[{1,1,-1}]

Out[]=
109.471
But:
In[]:=
180-109.47122063449069`
Out[]=
70.5288
Angle DarkGreen-Orange:
In[]:=
α
1
=N
180
π
ArcCos
{1,-1,1}.{1,1,1}
Norm[{1,-1,1}]Norm[{1,1,1}]

Out[]=
70.5288
Angle Purple-Brown:
In[]:=
α
1
=N
180
π
ArcCos
{1,1,-1}.{1,-1,-1}
Norm[{1,1,-1}]Norm[{1,-1,-1}]

Out[]=
70.5288
It is shown that the circles circumscribed to two faces of a tetrahedron are cut at the same angle as the circles circumscribed to the other two.I used a regular tetrahedron in this presentation, will it be fulfilled for an irregular tetrahedron?
In[]:=
gr=Show
ζ
1
[t_]:=-
1
3
,-
1
3
,-
1
3
+
2
2
3
Cos[t]
Norm[{-1,1,0}]
{-1,1,0}+
2
2
3
Sin[t]
Norm[{1,1,-2}]
{1,1,-2};
ζ
2
[t_]:=
1
3
,-
1
3
,
1
3
+
2
2
3
Cos[t]
Norm[{1,1,0}]
{1,1,0}+
2
2
3
Sin[t]
Norm[{1,-1,-2}]
{1,-1,-2};​​​​
ζ
3
[t_]:=-
1
3
,
1
3
,
1
3
+
2
2
3
Cos[t]
Norm[{1,1,0}]
{1,1,0}+
2
2
3
Sin[t]
Norm[{-1,1,-2}]
{-1,1,-2};​​​​
ζ
4
[t_]:=
1
3
,
1
3
,-
1
3
+
2
2
3
Cos[t]
Norm[{-1,1,0}]
{-1,1,0}+
2
2
3
Sin[t]
Norm[{-1,-1,-2}]
{-1,-1,-2};{ParametricPlot3D[
ζ
1
[t],{t,0,2π},PlotStyle{Orange,Thick}],​​ParametricPlot3D[
ζ
2
[t],{t,0,2π},PlotStyle{DarkGreen,Thick}],​​ParametricPlot3D[
ζ
3
[t],{t,0,2π},PlotStyle{Purple,Thick}],​​ParametricPlot3D[
ζ
4
[t],{t,0,2π},PlotStyle{Brown,Thick}],Graphics3D[{Black,Ball[{1,1,1},0.015],​​Text[Style[Subscript[P,1],14,Bold],{1,1,1},{0,-1.2}],​​Ball[{1,-1,-1},0.015],​​Text[Style[Subscript[P,2],14,Bold],{1,-1,-1},{0,-1.2}],​​Ball[{-1,1,-1},0.015],​​Text[Style[Subscript[P,3],14,Bold],{-1,1,-1},{0,-1.2}],​​Ball[{-1,-1,1},0.015],​​Text[Style[Subscript[P,4],14,Bold],{-1,-1,1},{0,-1.2}],Green,Opacity[0.1],EdgeForm[Thick],Tetrahedron[{{1,1,1},{1,-1,-1},{-1,1,-1},{-1,-1,1}}]}]},AxesLabel->{x,y,z},AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{1.3,-2.4,2.},ImageSize700