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Cyclic Hexagon

A hexagon (not necessarily regular) on whose Vertices a Circle may be Circumscribed. Let

\begin{displaymath}
\sigma_i\equiv \sum_{i,j,\dots,n=1} {a_i}^2 {a_j}^2 \cdots {a_n}^2,
\end{displaymath} (1)

where the sum runs over all distinct permutations of the Squares of the six side lengths, so
$\displaystyle \sigma_1$ $\textstyle =$ $\displaystyle {a_1}^2+{a_2}^2+{a_3}^2+{a_4}^2+{a_5}^2+{a_6}^2$ (2)
$\displaystyle \sigma_2$ $\textstyle =$ $\displaystyle {a_1}^2{a_2}^2+{a_1}^2{a_3}^2+{a_1}^2{a_4}^2+{a_1}^2{a_5}^2+{a_1}^2{a_6}^2$  
  $\textstyle \phantom{=}$ $\displaystyle +{a_2}^2{a_3}^2+{a_2}^2{a_4}^2+{a_2}^2{a_5}^2+{a_2}^2{a_6}^2$  
  $\textstyle \phantom{=}$ $\displaystyle +{a_3}^2{a_4}^2+{a_3}^2{a_5}^2+{a_3}^2{a_6}^2$  
  $\textstyle \phantom{=}$ $\displaystyle +{a_4}^2{a_5}^2+{a_4}^2{a_6}^2+{a_5}^2{a_6}^2$ (3)
$\displaystyle \sigma_3$ $\textstyle =$ $\displaystyle {a_1}^2{a_2}^2{a_3}^2+{a_1}^2{a_2}^2{a_4}^2+{a_1}^2{a_2}^2{a_5}^2+{a_1}^2{a_2}^2{a_6}^2$  
  $\textstyle \phantom{=}$ $\displaystyle +{a_2}^2{a_3}^2{a_4}^2+{a_2}^2{a_3}^2{a_5}^2+{a_2}^2{a_3}^2{a_6}^2$  
  $\textstyle \phantom{=}$ $\displaystyle +{a_3}^2{a_4}^2{a_5}^2+{a_3}^2{a_4}^2{a_6}^2+{a_4}^2{a_5}^2{a_6}^2$ (4)
$\displaystyle \sigma_4$ $\textstyle =$ $\displaystyle {a_1}^2{a_2}^2{a_3}^2{a_4}^2+{a_1}^2{a_2}^2{a_3}^2{a_5}^2+{a_1}^2{a_2}^2{a_3}^2{a_6}^2$  
  $\textstyle \phantom{=}$ $\displaystyle +{a_1}^2{a_3}^2{a_4}^2{a_5}^2+{a_1}^2{a_3}^2{a_4}^2{a_6}^2$  
  $\textstyle \phantom{=}$ $\displaystyle +{a_1}^2{a_3}^2{a_5}^2{a_6}^2+{a_1}^2{a_4}^2{a_5}^2{a_6}^2$  
  $\textstyle \phantom{=}$ $\displaystyle +{a_2}^2{a_3}^2{a_4}^2{a_5}^2+{a_2}^2{a_3}^2{a_4}^2{a_6}^2+{a_2}^2{a_3}^2{a_5}^2{a_6}^2$  
  $\textstyle \phantom{=}$ $\displaystyle +{a_2}^2{a_4}^2{a_5}^2{a_6}^2+{a_3}^2{a_4}^2{a_5}^2{a_6}^2$ (5)
$\displaystyle \sigma_5$ $\textstyle =$ $\displaystyle {a_1}^2{a_2}^2{a_3}^2{a_4}^2{a_5}^2+{a_1}^2{a_2}^2{a_3}^2{a_4}^2{a_6}^2$  
  $\textstyle \phantom{=}$ $\displaystyle +{a_1}^2{a_2}^2{a_3}^2{a_5}^2{a_6}^2+{a_1}^2{a_2}^2{a_4}^2{a_5}^2{a_6}^2$  
  $\textstyle \phantom{=}$ $\displaystyle +{a_1}^2{a_3}^2{a_4}^2{a_5}^2{a_6}^2+{a_2}^2{a_3}^2{a_4}^2{a_5}^2{a_6}^2$ (6)
$\displaystyle \sigma_6$ $\textstyle =$ $\displaystyle {a_1}^2{a_2}^2{a_3}^2{a_4}^2{a_5}^2{a_6}^2.$ (7)

Then define
$\displaystyle t_2$ $\textstyle =$ $\displaystyle u-4\sigma_2+{\sigma_1}^2$ (8)
$\displaystyle t_3$ $\textstyle =$ $\displaystyle 8\sigma_3+\sigma_1 t_2-16\sqrt{\sigma_6}$ (9)
$\displaystyle t_4$ $\textstyle =$ $\displaystyle {t_2}^2-64\sigma_4+64\sigma_1\sqrt{\sigma_6}$ (10)
$\displaystyle t_5$ $\textstyle =$ $\displaystyle 128\sigma_5+32t_2\sqrt{\sigma_6}$ (11)
$\displaystyle u$ $\textstyle =$ $\displaystyle 16K^2.$ (12)

The Area of the hexagon then satisfies
\begin{displaymath}
u{t_4}^3+{t_3}^2{t_4}^2-16{t_3}^3t_5-18ut_3t_4t_5-27u^2{t_5}^2=0,
\end{displaymath} (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 Discriminant of the Cubic Equation
\begin{displaymath}
z^3+2t_3z^2-ut_4 z+2y^2t_5.
\end{displaymath} (14)

See also Concyclic, Cyclic Pentagon, Cyclic Polygon, Fuhrmann's Theorem


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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© 1996-9 Eric W. Weisstein
1999-05-25