Cyclic Hexagon

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

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

where the sum runs over all distinct permutations of the Squares of the six side lengths, so

$$\sigma_1 = {a_1}^2+{a_2}^2+{a_3}^2+{a_4}^2+{a_5}^2+{a_6}^2$$ (2)

$$\sigma_2 = {a_1}^2{a_2}^2+{a_1}^2{a_3}^2+{a_1}^2{a_4}^2+{a_1}^2{a_5}^2+{a_1}^2{a_6}^2$$

$$\phantom{=} +{a_2}^2{a_3}^2+{a_2}^2{a_4}^2+{a_2}^2{a_5}^2+{a_2}^2{a_6}^2$$

$$\phantom{=} +{a_3}^2{a_4}^2+{a_3}^2{a_5}^2+{a_3}^2{a_6}^2$$

$$\phantom{=} +{a_4}^2{a_5}^2+{a_4}^2{a_6}^2+{a_5}^2{a_6}^2$$ (3)

$$\sigma_3 = {a_1}^2{a_2}^2{a_3}^2+{a_1}^2{a_2}^2{a_4}^2+{a_1}^2{a_2}^2{a_5}^2+{a_1}^2{a_2}^2{a_6}^2$$

$$\phantom{=} +{a_2}^2{a_3}^2{a_4}^2+{a_2}^2{a_3}^2{a_5}^2+{a_2}^2{a_3}^2{a_6}^2$$

$$\phantom{=} +{a_3}^2{a_4}^2{a_5}^2+{a_3}^2{a_4}^2{a_6}^2+{a_4}^2{a_5}^2{a_6}^2$$ (4)

$$\sigma_4 = {a_1}^2{a_2}^2{a_3}^2{a_4}^2+{a_1}^2{a_2}^2{a_3}^2{a_5}^2+{a_1}^2{a_2}^2{a_3}^2{a_6}^2$$

$$\phantom{=} +{a_1}^2{a_3}^2{a_4}^2{a_5}^2+{a_1}^2{a_3}^2{a_4}^2{a_6}^2$$

$$\phantom{=} +{a_1}^2{a_3}^2{a_5}^2{a_6}^2+{a_1}^2{a_4}^2{a_5}^2{a_6}^2$$

$$\phantom{=} +{a_2}^2{a_3}^2{a_4}^2{a_5}^2+{a_2}^2{a_3}^2{a_4}^2{a_6}^2+{a_2}^2{a_3}^2{a_5}^2{a_6}^2$$

$$\phantom{=} +{a_2}^2{a_4}^2{a_5}^2{a_6}^2+{a_3}^2{a_4}^2{a_5}^2{a_6}^2$$ (5)

$$\sigma_5 = {a_1}^2{a_2}^2{a_3}^2{a_4}^2{a_5}^2+{a_1}^2{a_2}^2{a_3}^2{a_4}^2{a_6}^2$$

$$\phantom{=} +{a_1}^2{a_2}^2{a_3}^2{a_5}^2{a_6}^2+{a_1}^2{a_2}^2{a_4}^2{a_5}^2{a_6}^2$$

$$\phantom{=} +{a_1}^2{a_3}^2{a_4}^2{a_5}^2{a_6}^2+{a_2}^2{a_3}^2{a_4}^2{a_5}^2{a_6}^2$$ (6)

$$\sigma_6 = {a_1}^2{a_2}^2{a_3}^2{a_4}^2{a_5}^2{a_6}^2.$$ (7)

Then define

$$t_2 = u-4\sigma_2+{\sigma_1}^2$$ (8)

$$t_3 = 8\sigma_3+\sigma_1 t_2-16\sqrt{\sigma_6}$$ (9)

$$t_4 = {t_2}^2-64\sigma_4+64\sigma_1\sqrt{\sigma_6}$$ (10)

$$t_5 = 128\sigma_5+32t_2\sqrt{\sigma_6}$$ (11)

$$u = 16K^2.$$ (12)

The Area of the hexagon then satisfies

$$u{t_4}^3+{t_3}^2{t_4}^2-16{t_3}^3t_5-18ut_3t_4t_5-27u^2{t_5}^2=0,$$ (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

$$z^3+2t_3z^2-ut_4 z+2y^2t_5.$$ (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.