In[]:=
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=4;d=-3;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


5
3
,6,
47
9
,6+
32
3
3
,
151
33
,6+
32
11
3
,
203
45
,6,
151
33
,6-
32
11
3
,
47
9
,6-
32
3
3
,
5
3
,6;
In[]:=
​​​​​​Showr=2;k=4;d=-3;n=6;ρ=
2Sin
π
n

1-Sin
π
n

;x1=4;y1=6;​​ParametricPlot
128
90
Cosh[t]+
139
45
,
64
3
15
Sinh[t]+6,{t,-π,π},PlotStyle->{Black,Thick},ParametricPlot-
128
90
Cosh[t]+
139
45
,
64
3
15
Sinh[t]+6,{t,-π,π},PlotStyle->{Black,Thick},​​TableGraphicsOrange,Thickness[0.01],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
,GraphicsRed,PointSize[0.018],Point
5
3
,6,GraphicsBlue,PointSize[0.018],Point
47
9
,6+
32
3
3
,GraphicsGreen,PointSize[0.018],Point
151
33
,6+
32
11
3
,GraphicsGray,PointSize[0.018],Point
203
45
,6,GraphicsPink,PointSize[0.018],Point
151
33
,6-
32
11
3
,GraphicsPurple,PointSize[0.018],Point
47
9
,6-
32
3
3
,AxesTrue,BoxRatiosAutomatic,AxesOrigin{0,0},ImageSize700,PlotRange{{0,10},{0,15}}
Out[]=
In[]:=
​​​​​​Showr=2;k=4;d=-3;n=6;ρ=
2Sin
π
n

1-Sin
π
n

;x1=4;y1=6;​​TableGraphicsBlack,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
,GraphicsRed,PointSize[0.018],Point
5
3
,6,GraphicsBlue,PointSize[0.018],Point
47
9
,6+
32
3
3
,GraphicsGreen,PointSize[0.018],Point
151
33
,6+
32
11
3
,GraphicsGray,PointSize[0.018],Point
203
45
,6,GraphicsPink,PointSize[0.018],Point
151
33
,6-
32
11
3
,GraphicsPurple,PointSize[0.018],Point
47
9
,6-
32
3
3
,AxesTrue,BoxRatiosAutomatic,AxesOrigin{0,0},ImageSize700,PlotRange{{0,10},{-1,15}}
Obtaining the equations of the opposite sides:
PurplePink line and BlueGreen, form the middle point between Red and Gray points:
Complete graph:
The graph shows Pascal’s line.