  The B\^{o}cher-Grace Theorem can be stated as follows: Let $p$ be a third
degree complex polynomial. Then there is a unique inscribed ellipse
interpolating the midpoints of the triangle formed from the roots of $p$, and
the foci of the ellipse are the critical points of $p$. Here, we prove the
following generalization: Let $p$ be an $n^{th}$ degree complex polynomial and
let its critical points take the form $$ \alpha+\beta \cos k\pi/n, \quad
k=1,...,n-1, \quad\beta\ne0. $$ Then there is an inscribed ellipse
interpolating the midpoints of the convex polygon formed by the roots of $p$,
and the foci of this ellipse are the two most extreme critical points of $p$:
$\alpha\pm\beta \cos \pi/n$.
