In[]:=
BrianchonTheorem::usage="It states that, given a hexagon circumscribed on a conic section, the lines joining opposite polygon vertices (polygon diagonals) meet in single point.Reference:WolframMathWorld:Brianchon'sTheorem;Personal";
The following animation verifies the theorem:
In[]:=
​​Manipulate​​Showr=2;k=4;d=-2;n=6;ρ=
2Sin
π
n

1-Sin
π
n

;x1=4;y1=6;​​TableGraphicsBlack,Dashed,Thick,InfiniteLine{4-d,y1}+
2
k
2
((r+ρ)Cos[t1+2πk1]+x1-(4-d))
+
2
((r+ρ)Sin[t1+2πk1]+y1-6)
-
2
ρ
{(r+ρ)Cos[t1+2πk1]+x1-(4-d),(r+ρ)Sin[t1+2πk1]+y1-6},{4-d,y1}+
2
k
2
((r+ρ)Cos[t1+2πk1+π]+x1-(4-d))
+
2
((r+ρ)Sin[t1+2πk1+π]+y1-6)
-
2
ρ
{(r+ρ)Cos[t1+2πk1+π]+x1-(4-d),(r+ρ)Sin[t1+2πk1+π]+y1-6},t1,0,2π,
2π
n
,​​TableGraphicsOrange,PointSize[0.015],Point{4-d,y1}+
2
k
2
((r+ρ)Cos[t1+2πk1]+x1-(4-d))
+
2
((r+ρ)Sin[t1+2πk1]+y1-6)
-
2
ρ
{(r+ρ)Cos[t1+2πk1]+x1-(4-d),(r+ρ)Sin[t1+2πk1]+y1-6},t1,0,2π,
2π
n
,​​​​TableGraphicsOrange,Thick,Line{4-d,y1}+
2
k
2
((r+ρ)Cos[t1+2πk1]+x1-(4-d))
+
2
((r+ρ)Sin[t1+2πk1]+y1-6)
-
2
ρ
{(r+ρ)Cos[t1+2πk1]+x1-(4-d),(r+ρ)Sin[t1+2πk1]+y1-6},{4-d,y1}+
2
k

2
(r+ρ)Cost1+
2π
n
+2πk1+x1-(4-d)
+
2
(r+ρ)Sint1+
2π
n
+2πk1+y1-6
-
2
ρ
(r+ρ)Cost1+
2π
n
+2πk1+x1-(4-d),(r+ρ)Sint1+
2π
n
+2πk1+y1-6,t1,0,2π,
2π
n
,​​​​ContourPlot[
2
y
-16x-12y+84==0,{x,0,40},{y,-20,40},ContourStyle->{Black,Thick},PlotPoints->50],Graphics[{Black,PointSize[0.015],Point[{4,6}]}],AxesTrue,AxesLabel{x,y},BoxRatiosAutomatic,AxesOrigin{0,0},ImageSize700,PlotRange{{0,30},{-20,30}},Style["Brianchon's Theorem:",Bold,Large],{{k1,0.00002,"Valor (k1)"},0.00001,6,0.2},ControlPlacementTop
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​
Brianchon's Theorem:
Valor (k1)