Testing whether the circumcenter of a cyclic quadrilateral lies inside it For a triangle with sides $a, b, c$ (where $c$ is the biggest side) there is a simple check to see whether it's circumcenter lies inside of it:
$$a^2 + b^2 < c^2$$
Is there such an inequality for a cyclic quadrilateral, given its side lengths $a, b, c, d$ (with longest side $d$)?
Can this be generalized to a cyclic convex $n$-gon?

The circumcenter of a triangle is inside the triangle if and only if
$$
(a^2+b^2-c^2)(c^2+a^2-b^2)(b^2+c^2-a^2)\gt0\tag{1}
$$

The diagonal with sides $a$ and $b$ on  one side and $c$ and $d$ on the other is
$$
e^2=\frac{\frac{a^2+b^2}{ab}+\frac{c^2+d^2}{cd}}{\frac1{ab}+\frac1{cd}}\tag{2}
$$
Then the circumcenter is inside the quadrilateral if and only if
$$
\hspace{-10pt}\small(a^2+b^2-e^2)(e^2+a^2-b^2)(b^2+e^2-a^2)(c^2+d^2-e^2)(e^2+c^2-d^2)(d^2+e^2-c^2)\lt0\tag{3}
$$
or
$$
a^2+b^2=c^2+d^2\tag{4}
$$
Note that $(4)$ implies equality in $(3)$.
