$A$ line drawn parallel to $\overline{YZ}$ in the plane of $\Delta XYZ$ passes through the midpoint of $\overline{XY}$. Prove that this line bisects $\overline{XZ}$.

(N/A) Let $M$ be the midpoint of $\overline{XY}$ such that $XM = MY$. Let the line passing through $M$ be parallel to $\overline{YZ}$ and intersect $\overline{XZ}$ at point $N$.
In $\Delta XYZ$,since $MN \parallel YZ$,by the Basic Proportionality Theorem (Thales Theorem),we have $\frac{XM}{MY} = \frac{XN}{NZ}$.
Since $M$ is the midpoint of $\overline{XY}$,$XM = MY$,which implies $\frac{XM}{MY} = 1$.
Substituting this into the ratio,we get $1 = \frac{XN}{NZ}$,which means $XN = NZ$.
Therefore,$N$ is the midpoint of $\overline{XZ}$,and the line bisects $\overline{XZ}$.