Cyclic Pentagon

Author

Eric W. Weisstein
October 18, 2005

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

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

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

Initialization

Discriminant[p_?PolynomialQ,x_]:=With[{m=Exponent[p,x]},Cancel[((-1)^(1/2m(m-1))Resultant[p,D[p,x],x])/Coefficient[p,x,m]^(2m-1)]
]

Symmetric Polynomials

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

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

Check

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;

Area[n_]:=FullSimplify[1/4 n Cot[Pi/n]]

cond=Append[Thread[Table[a[n],{n,5}]->Table[1,{5}]] , K->Area[5]]

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]

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

Simplify[disc /. cond]

Simplify[eqn /. cond]