<s>[INST] Cyclic Hexagon Circumradius A cyclic hexagon has side lengths of 2, 2, 7, 7, 11, 11, in that order. Find the length of its circumradius.
Not sure if there is a theorem or formula for this, but I tried dividing it into 30°, 60°, 90° triangles. Is that a possible way to approach the problem? If there is a theorem that can be used for this I would love to know. Any help is greatly appreciated.
 [/INST] In general would be the solution for
$4 R^3 -(a^2+b^2+c^2) R - a b c = 0$
$P = \sqrt[3]{ \sqrt{4(-12)^3(a^2+b^2+c^2)^3 + 2^8 3^6 a^2 b^2 c^2} + 2^4 3^3 a b c }$
$
R = \frac{P}{12 \sqrt[3]{2}} + \frac{\sqrt[3]{2} (a^{2}+b^{2}+c^{2})}{P}
$
then
$a=2, b=7, c=11$
$4 R^3 - 174 R - 154 = 0$
$R = 7$
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