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Cyclic Pentagon

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A cyclic pentagon is a not necessarily regular pentagon on whose polygon vertices a circle may be circumscribed. Let such a pentagon have edge lengths a_1, ..., a_5, and area K, and let

sigma_i=Pi_i(a_1^2,a_2^2,a_3^2,a_4^2,a_5^2)
(1)

denote the ith-order symmetric polynomial on the five variables consisting of the squares a_i^2 of the pentagon side lengths a_i, so

sigma_1=a_1^2+a_2^2+a_3^2+a_4^2+a_5^2
(2)
sigma_2=a_1^2a_2^2+a_1^2a_3^2+a_1^2a_4^2+a_1^2a_5^2+a_2^2a_3^2+a_2^2a_4^2+a_2^2a_5^2+a_3^2a_4^2+a_3^2a_5^2+a_4^2a_5^2
(3)
sigma_3=a_1^2a_2^2a_3^2+a_1^2a_2^2a_4^2+a_1^2a_2^2a_5^2+a_1^2a_3^2a_4^2+a_1^2a_3^2a_5^2+a_1^2a_4^2a_5^2+a_2^2a_3^2a_4^2+a_2^2a_3^2a_5^2+a_2^2a_4^2a_5^2+a_3^2a_4^2a_5^2
(4)
sigma_4=a_1^2a_2^2a_3^2a_4^2+a_1^2a_2^2a_3^2a_5^2+a_1^2a_3^2a_4^2a_5^2+a_1^2a_2^2a_4^2a_5^2+a_2^2a_3^2a_4^2a_5^2
(5)
sigma_5=a_1^2a_2^2a_3^2a_4^2a_5^2.
(6)

In addition, also define

u=16K^2
(7)
t_2=u-4sigma_2+sigma_1^2
(8)
t_3=8sigma_3+sigma_1t_2
(9)
t_4=-64sigma_4+t_2^2
(10)
t_5=128sigma_5.
(11)

Then the area of the pentagon satisfies

ut_4^3+t_3^2t_4^2-16t_3^3t_5-18ut_3t_4t_5-27u^2t_5^2=0,
(12)

a seventh order polynomial in u (Robbins 1995). This is also 1/(4u^2) times the polynomial discriminant of the cubic equation

z^3+2t_3z^2-ut_4z+2u^2t_5
(13)

(Robbins 1995).

See also

Concyclic, Cyclic Hexagon, Cyclic Polygon, Cyclic Quadrilateral

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References

Robbins, D. P. "Areas of Polygons Inscribed in a Circle." Discr. Comput. Geom. 12, 223-236, 1994.Robbins, D. P. "Areas of Polygons Inscribed in a Circle." Amer. Math. Monthly 102, 523-530, 1995.

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Cyclic Pentagon

Cite this as:

Weisstein, Eric W. "Cyclic Pentagon." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CyclicPentagon.html

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