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Cyclic Hexagon

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A hexagon (not necessarily regular) on whose polygon vertices a circle may be circumscribed. Let

sigma_i=Pi_i(a_1^2,a_2^2,a_3^2,a_4^2,a_5^2,a_6^2)
(1)

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

sigma_1=a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2
(2)
sigma_2=a_1^2a_2^2+a_1^2a_3^2+a_1^2a_4^2+a_1^2a_5^2+a_1^2a_6^2+a_2^2a_3^2+a_2^2a_4^2+a_2^2a_5^2+a_2^2a_6^2+a_3^2a_4^2+a_3^2a_5^2+a_3^2a_6^2+a_4^2a_5^2+a_4^2a_6^2+a_5^2a_6^2
(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_2^2a_6^2+a_2^2a_3^2a_4^2+a_2^2a_3^2a_5^2+a_2^2a_3^2a_6^2+a_3^2a_4^2a_5^2+a_3^2a_4^2a_6^2+a_4^2a_5^2a_6^2
(4)
sigma_4=a_1^2a_2^2a_3^2a_4^2+a_1^2a_2^2a_3^2a_5^2+a_1^2a_2^2a_3^2a_6^2+a_1^2a_3^2a_4^2a_5^2+a_1^2a_3^2a_4^2a_6^2+a_1^2a_3^2a_5^2a_6^2+a_1^2a_4^2a_5^2a_6^2+a_2^2a_3^2a_4^2a_5^2+a_2^2a_3^2a_4^2a_6^2+a_2^2a_3^2a_5^2a_6^2+a_2^2a_4^2a_5^2a_6^2+a_3^2a_4^2a_5^2a_6^2
(5)
sigma_5=a_1^2a_2^2a_3^2a_4^2a_5^2+a_1^2a_2^2a_3^2a_4^2a_6^2+a_1^2a_2^2a_3^2a_5^2a_6^2+a_1^2a_2^2a_4^2a_5^2a_6^2+a_1^2a_3^2a_4^2a_5^2a_6^2+a_2^2a_3^2a_4^2a_5^2a_6^2
(6)
sigma_6=a_1^2a_2^2a_3^2a_4^2a_5^2a_6^2.
(7)

Then let K be the area of the hexagon and define

u=16K^2
(8)
t_2=u-4sigma_2+sigma_1^2
(9)
t_3=8sigma_3+sigma_1t_2-16sqrt(sigma_6)
(10)
t_4=t_2^2-64sigma_4+64sigma_1sqrt(sigma_6)
(11)
t_5=128sigma_5+32t_2sqrt(sigma_6).
(12)

The area of the hexagon then satisfies

ut_4^3+t_3^2t_4^2-16t_3^3t_5-18ut_3t_4t_5-27u^2t_5^2=0,
(13)

or this equation with sqrt(sigma_6) replaced by -sqrt(sigma_6), a seventh-order polynomial in u. This is 1/(4u^2) times the polynomial discriminant of the cubic equation

z^3+2t_3z^2-ut_4z+2u^2t_5.
(14)

See also

Concyclic, Cyclic Pentagon, Cyclic Polygon, Cyclic Quadrilateral, Fuhrmann's Theorem, Lemoine Hexagon

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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.

Referenced on Wolfram|Alpha

Cyclic Hexagon

Cite this as:

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

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