Cyclic Hexagon

Eric W. Weisstein
January 11, 2002

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/CyclicHexagon.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/CyclicHexagon.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Construction

σ[n_,k_:5]:=Plus@@Times@@@(Map[a,KSubsets[Range[k],n],{2}]^2)

Table[σ[n],{n,5}]/.a[n_]:>Subscript[a,n]//ColumnForm//TraditionalForm

u=16K^2;
(Subscript[t, 2])=u-4σ[2,6]+σ[1,6]^2;
(Subscript[t, 3])=8σ[3,6]+σ[1,6](Subscript[t, 2])+16Sqrt[σ[6,6]];
(Subscript[t, 4])=(Subscript[t, 2])^2-64σ[4,6]-64σ[1,6]Sqrt[σ[6,6]];
(Subscript[t, 5])=128σ[5,6]-32(Subscript[t, 2])Sqrt[σ[6,6]];

{u,(Subscript[t, 2]),(Subscript[t, 3]),(Subscript[t, 4]),(Subscript[t, 5])}=.

cond=Append[Thread[Table[a[n],{n,6}]->Table[1,{6}]] , K->Area[6]]

eqn=u (Subscript[t, 4])^3+(Subscript[t, 3])^2 (Subscript[t, 4])^2-16(Subscript[t, 3])^3 (Subscript[t, 5]) -18u (Subscript[t, 3 ])(Subscript[t, 4]) (Subscript[t, 5])-27u^2 (Subscript[t, 5])^2

disc=Expand[Discriminant[z^3+2(Subscript[t, 3])z^2-u (Subscript[t, 4])z+2u^2 (Subscript[t, 5]),z]/4/u^2]

Simplify[disc /. cond]

Simplify[eqn /. cond]