DesarguesTheorem::usage="Given the triangles in space: A{-2, 3, 5}​, B{5, 2, 5}​, C{3, 7, 5} and A'-
10
17
​,​
57
17
​,​
175
17
​​, B'​
40
9
​,​
64
27
​,​
70
9
​​, C'​
80
29
​,​
182
29
​,​
250
29
​: Find the perspector (point where the lines that join the corresponding vertices in both triangles are found) and the perspectrix (line formed by the three intersections of the corresponding pairs of sides). Reference: Personal, based in Desargues' Theorem.";
In[]:=
Showtri1=Polygon[{{-2,3,5},{5,2,5},{3,7,5}}];tri2=Polygon-
10
17
,
57
17
,
175
17
,
40
9
,
64
27
,
70
9
,
80
29
,
182
29
,
250
29
;Graphics3D[{EdgeForm[Thick],Green,Opacity[0.1],tri1}],Graphics3D[{EdgeForm[Thick],Yellow,Opacity[0.1],tri2}],Graphics3D[{Black,Ball[{-2,3,5},0.07]}],Graphics3D[{Red,Ball[{5,2,5},0.07]}],Graphics3D[{LightGray,Ball[{3,7,5},0.07]}],Graphics3DBlack,Ball-
10
17
,
57
17
,
175
17
,0.07,Graphics3DRed,Ball
40
9
,
64
27
,
70
9
,0.07,Graphics3DLightGray,Ball
80
29
,
182
29
,
250
29
,0.07,Graphics3DBlack,Thick,Dashed,InfiniteLine{-2,3,5},-
10
17
,
57
17
,
175
17
,Graphics3DBlack,Thick,Dashed,InfiniteLine{5,2,5},
40
9
,
64
27
,
70
9
,Graphics3DBlack,Thick,Dashed,InfiniteLine{3,7,5},
80
29
,
182
29
,
250
29
,AxesLabel->{x,y,z},AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{5,-2.4,1.},ImageSize700
Out[]=
Development:
Perspector:
Black points line:
In[]:=
-
10
17
,
57
17
,
175
17
-{-2,3,5}

24
17
,
6
17
,
90
17
;
Red points line:
In[]:=

40
9
,
64
27
,
70
9
-{5,2,5}
-
5
9
,
10
27
,
25
9
;
In[]:=
Solve
x+2
24
17
==
y-3
6
17
==
z-5
90
17
,
x-5
-
5
9
==
y-2
10
27
==
z-5
25
9
,{x,y,z}
{{x2,y4,z20}};
Now it is replaced in the LightGray points line to verify that the three straight lines intersect at that point:
In[]:=

80
29
,
182
29
,
250
29
-{3,7,5}
-
7
29
,-
21
29
,
105
29
;
In[]:=
2-3
-
7
29
==
4-7
-
21
29
==
20-5
105
29
Out[]=
True
Therefore the perspector is:
{2,4,20};
Perspectrix:
Top Black-Red points line:
In[]:=

40
9
,
64
27
,
70
9
--
10
17
,
57
17
,
175
17


770
153
,-
451
459
,-
385
153
;
Bottom Black-Red points line:
In[]:=
{5,2,5}-{-2,3,5}
{7,-1,0};
In[]:=
Solve
x+
10
17
770
153
==
y-
57
17
-
451
459
==
z-
175
17
-
385
153
,
x+2
7
==
y-3
-1
,z==5,{x,y,z}
x10,y
9
7
,z5;
Top LightGray-Red points line:
In[]:=

40
9
,
64
27
,
70
9
-
80
29
,
182
29
,
250
29


440
261
,-
3058
783
,-
220
261
;
Bottom LightGray-Red points line:
In[]:=
{5,2,5}-{3,7,5}
{2,-5,0};
In[]:=
Solve
x-
80
29
440
261
==
y-
182
29
-
3058
783
==
z-
250
29
-
220
261
,
x-3
2
==
y-7
-5
,z==5,{x,y,z}
x10,y-
21
2
,z5;
Top Black-LightGray points line:
In[]:=

80
29
,
182
29
,
250
29
--
10
17
,
57
17
,
175
17


1650
493
,
1441
493
,-
825
493
;
Bottom Black-LightGray points line:
The perspectrix equations:
Verifying with the graph:
It is verified that the three points are colinear.
Full graph: