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:
P1={1,1,1};P2={3,6,0};P3={0,3,2};P4={2,3,7};
Cartesian equation of the
P
2
P
3
P
4
plane:
In[]:=
P4={2,3,7};P3={0,3,2};P2={3,6,0};​​P4-P2​​P3-P2
{-1,-3,7};
{-3,-3,2};
In[]:=
{-1,-3,7}{-3,-3,2}
{15,-19,-6};
15x-19y-6z+D==0;​​15(0)-19(3)-6(2)+D==0;​​D=69;​​15x-19y-6z+69==0;
In[]:=
Solve[{15x-19y-6z+69==0,
2
(x-2)
+
2
(y-3)
+
2
(z-7)
==
2
(x-0)
+
2
(y-3)
+
2
(z-2)
,
2
(x-2)
+
2
(y-3)
+
2
(z-7)
==
2
(x-3)
+
2
(y-6)
+
2
(z-0)
},{x,y,z}]
x
1857
622
,y
2997
622
,z
2305
622
;
In[]:=
x=
19
15
y+
2
5
z-
69
15
;​​y=u;​​z=v;​​
19
15
,1,0{15,-19,-6}
-6,
38
5
,-
586
15
;
In[]:=
​​r1=Norm
1857
622
,
2997
622
,
2305
622
-{0,3,2}
18821
311
2
;
ζ
1
[t_]:=
1857
622
,
2997
622
,
2305
622
+
18821
311
2
Cos[t]
Norm
19
15
,1,0

19
15
,1,0+
18821
311
2
Sin[t]
Norm-6,
38
5
,-
586
15

-6,
38
5
,-
586
15
;
Cartesian equation of the
P
1
P
2
P
4
plane:
In[]:=
P1={1,1,1};P2={3,6,0};P4={2,3,7};​​P4-P1​​P2-P1​​
{1,2,6};
{2,5,-1};
In[]:=
{1,2,6}{2,5,-1}
{-32,13,1};
-32x+13y+z+D=0;​​-32(1)+13(1)+1+D=0;​​D=18;​​32x-13y-z-18==0;
In[]:=
Solve[{32x-13y-z-18==0,
2
(x-1)
+
2
(y-1)
+
2
(z-1)
==
2
(x-3)
+
2
(y-6)
+
2
(z-0)
,
2
(x-1)
+
2
(y-1)
+
2
(z-1)
==
2
(x-2)
+
2
(y-3)
+
2
(z-7)
},{x,y,z}]
x
901
398
,y
784
199
,z
642
199
;
In[]:=
​​x=
13
32
y+
1
32
z+
18
32
;​​y=u;z=v;​​
13
32
,1,0{32,-13,-1}
-1,
13
32
,-
1193
32
;
In[]:=
r2=Norm
901
398
,
784
199
,
642
199
-{1,1,1}
12095
199
2
;
ζ
2
[t_]:=
901
398
,
784
199
,
642
199
+
12095
199
2
Cos[t]
Norm
13
32
,1,0

13
32
,1,0+
12095
199
2
Sin[t]
Norm-1,
13
32
,-
1193
32

-1,
13
32
,-
1193
32
;
Cartesian equation of the
P
1
P
3
P
4
plane:
In[]:=
P4={2,3,7};P1={1,1,1};P3={0,3,2};​​P4-P1​​P3-P1​​
{1,2,6};
{-1,2,1};
In[]:=
{1,2,6}{-1,2,1}
{-10,-7,4};
​​-10x-7y+4z+D=0;​​-10(1)-7(1)+4(1)+D=0;​​D=13;​​10x+7y-4z-13==0;
In[]:=
Solve[{10x+7y-4z-13==0,
2
(x-1)
+
2
(y-1)
+
2
(z-1)
==
2
(x-2)
+
2
(y-3)
+
2
(z-7)
,
2
(x-1)
+
2
(y-1)
+
2
(z-1)
==
2
(x-0)
+
2
(y-3)
+
2
(z-2)
},{x,y,z}]
x
43
22
,y
78
55
,z
453
110
;
In[]:=
x=-
7
10
y+
2
5
z+
13
10
;​​y=u;​​z=v;​​-
7
10
,1,0{10,7,-4}
-4,-
14
5
,-
149
10
;
In[]:=
​​r3=Norm
43
22
,
78
55
,
453
110
-{1,1,1}
1189
110
;
ζ
3
[t_]:=
43
22
,
78
55
,
453
110
+
1189
110
Cos[t]
Norm-
7
10
,1,0
-
7
10
,1,0+
1189
110
Sin[t]
Norm-4,-
14
5
,-
149
10

-4,-
14
5
,-
149
10
;
Cartesian equation of the
P
1
P
2
P
3
plane:
In[]:=
P2={3,6,0};P1={1,1,1};P3={0,3,2};​​P3-P1​​P2-P1
{-1,2,1};
{2,5,-1};
In[]:=
{-1,2,1}{2,5,-1}
{-7,1,-9};
Angle Orange-Purple:
Angle DarkGreen-Brown:
Angle DarkGreen-Orange:
Angle Purple-Brown: