Let $μ_1$ be a complex number in the numerical range $W(A)$ of a normal matrix $A$. In the case when no eigenvalues of $A$ lie in the interior of $W(A)$, we identify the smallest convex region containing all possible complex numbers $μ_2$ for which $\begin{bmatrix}μ_1& *\\0& μ_2\end{bmatrix}$ is a $2$-by-$2$ compression of $A$.
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