PascalTheorem::usage=" In any hexagon inscribed in a conical section, the continuations of opposite sides meet on a straight line, called the Pascal line. This property is repeated in Geometry, there is also the Monge's theorem and the Desargues' theorem. References: Personal, based in the Pascal's theorem. ";
Development:
Obtaining the vertices of the hexagon in numerical form:
In[]:=
r=2;k=2;d=1;n=6;ρ=
2Sin
π
n

1-Sin
π
n

;x1=4;y1=6;​​Table{4-d,y1}+
2
k
2
((r+ρ)Cos[t1]+x1-(4-d))
+
2
((r+ρ)Sin[t1]+y1-6)
-
2
ρ
{(r+ρ)Cos[t1]+x1-(4-d),(r+ρ)Sin[t1]+y1-6},t1,0,2π,
2π
n


83
21
,6,
63
17
,6+
8
3
17
,
23
9
,6+
8
3
3
,
3
5
,6,
23
9
,6-
8
3
3
,
63
17
,6-
8
3
17
,
83
21
,6;
In[]:=
​​​​​​Showr=2;k=2;d=1;n=6;ρ=
2Sin
π
n

1-Sin
π
n

;x1=4;y1=6;​​GraphicsRed,PointSize[0.015],Point
83
21
,6,GraphicsBlue,PointSize[0.015],Point
63
17
,6+
8
3
17
,GraphicsGreen,PointSize[0.015],Point
23
9
,6+
8
3
3
,GraphicsGray,PointSize[0.015],Point
3
5
,6,GraphicsPink,PointSize[0.015],Point
23
9
,6-
8
3
3
,GraphicsPurple,PointSize[0.015],Point
63
17
,6-
8
3
17
,​​ParametricPlot
176
105
Cos[θ]+
239
105
,
16
105
Sin[θ]+6,{θ,0,2π},ColorFunction->"DarkRainbow",​​TableGraphicsDarkGreen,Thick,Line{4-d,y1}+
2
k
2
((r+ρ)Cos[t1]+x1-(4-d))
+
2
((r+ρ)Sin[t1]+y1-6)
-
2
ρ
{(r+ρ)Cos[t1]+x1-(4-d),(r+ρ)Sin[t1]+y1-6},{4-d,y1}+
2
k

2
(r+ρ)Cost1+
2π
n
+x1-(4-d)
+
2
(r+ρ)Sint1+
2π
n
+y1-6
-
2
ρ
(r+ρ)Cost1+
2π
n
+x1-(4-d),(r+ρ)Sint1+
2π
n
+y1-6,t1,0,2π,
2π
n
,TableGraphicsBlack,Thick,Dashed,InfiniteLine{4-d,y1}+
2
k
2
((r+ρ)Cos[t1]+x1-(4-d))
+
2
((r+ρ)Sin[t1]+y1-6)
-
2
ρ
{(r+ρ)Cos[t1]+x1-(4-d),(r+ρ)Sin[t1]+y1-6},{4-d,y1}+
2
k

2
(r+ρ)Cost1+
2π
n
+x1-(4-d)
+
2
(r+ρ)Sint1+
2π
n
+y1-6
-
2
ρ
(r+ρ)Cost1+
2π
n
+x1-(4-d),(r+ρ)Sint1+
2π
n
+y1-6,t1,0,2π,
2π
n
,AxesTrue,BoxRatiosAutomatic,AxesOrigin{0,0},ImageSize700,PlotRange{{0,10},{0,10}}
Out[]=
Obtaining the equations of the opposite sides:
GrayGreen line and PurpleRedline, form the point 1:
In[]:=
Solve(y-6)
23
9
-
3
5
==6+
8
3
3
-6x-
3
5
,y-6-
8
3
17
83
21
-
63
17
==6-6-
8
3
17
x-
63
17
,{x,y}
x5,y23+
3
;
GreenBlue line and PinkPurple, form the point 2:
In[]:=
Solvey-6+
8
3
3
63
17
-
23
9
==6+
8
3
17
-6+
8
3
3
x-
23
9
,y-6-
8
3
3
63
17
-
23
9
==6-
8
3
17
-6-
8
3
3
x-
23
9
,{x,y}
{{x5,y6}};
GrayPink line and BlueRed line, form the point 3:
In[]:=
Solve(y-6)
23
9
-
3
5
==6-
8
3
3
-6x-
3
5
,y-6+
8
3
17
83
21
-
63
17
==6-6+
8
3
17
x-
63
17
,{x,y}
x5,y-2-3+
3
;
Complete graph:
In[]:=
​​​​​​Showr=2;k=2;d=1;n=6;ρ=
2Sin
π
n

1-Sin
π
n

;x1=4;y1=6;​​GraphicsRed,PointSize[0.015],Point
83
21
,6,GraphicsBlue,PointSize[0.015],Point
63
17
,6+
8
3
17
,GraphicsGreen,PointSize[0.015],Point
23
9
,6+
8
3
3
,GraphicsGray,PointSize[0.015],Point
3
5
,6,GraphicsPink,PointSize[0.015],Point
23
9
,6-
8
3
3
,GraphicsPurple,PointSize[0.015],Point
63
17
,6-
8
3
17
,​​Graphics[Black,PointSize[0.015],Point[5,23+
3
]],Graphics[{Black,PointSize[0.015],Point[{5,6}]}],Graphics[Black,PointSize[0.015],Point[5,-2-3+
3
]],Graphics[Black,Thick,InfiniteLine[{5,6},5,-2-3+
3
]],​​ParametricPlot
176
105
Cos[θ]+
239
105
,
16
105
Sin[θ]+6,{θ,0,2π},ColorFunction->"DarkRainbow",​​TableGraphicsDarkGreen,Thick,Line{4-d,y1}+
2
k
2
((r+ρ)Cos[t1]+x1-(4-d))
+
2
((r+ρ)Sin[t1]+y1-6)
-
2
ρ
{(r+ρ)Cos[t1]+x1-(4-d),(r+ρ)Sin[t1]+y1-6},{4-d,y1}+
2
k

2
(r+ρ)Cost1+
2π
n
+x1-(4-d)
+
2
(r+ρ)Sint1+
2π
n
+y1-6
-
2
ρ
(r+ρ)Cost1+
2π
n
+x1-(4-d),(r+ρ)Sint1+
2π
n
+y1-6,t1,0,2π,
2π
n
,TableGraphicsBlack,Thick,Dashed,InfiniteLine{4-d,y1}+
2
k
2
((r+ρ)Cos[t1]+x1-(4-d))
+
2
((r+ρ)Sin[t1]+y1-6)
-
2
ρ
{(r+ρ)Cos[t1]+x1-(4-d),(r+ρ)Sin[t1]+y1-6},{4-d,y1}+
2
k

2
(r+ρ)Cost1+
2π
n
+x1-(4-d)
+
2
(r+ρ)Sint1+
2π
n
+y1-6
-
2
ρ
(r+ρ)Cost1+
2π
n
+x1-(4-d),(r+ρ)Sint1+
2π
n
+y1-6,t1,0,2π,
2π
n
,AxesTrue,BoxRatiosAutomatic,AxesOrigin{0,0},ImageSize700,PlotRange{{0,10},{0,10}}
The graph shows Pascal’s line.