Let $P$ be an interior point of circle $K$ other than the center of $K$. Form all chords of $K$ which pass through $P$, and determine their midpoints. The locus of these midpoints is
$\textbf{(A)} \text{ a circle with one point deleted}$$\\ \textbf{(B)} \text{ a circle if the distance from } P \text{ to the center of } K \text{ is less than one half the radius of } K; \\ \text{otherwise a circular arc of less than } 360^{\circ}$$\\ \textbf{(C)} \text{ a semicircle with one point deleted}$$\\ \textbf{(D)} \text{ a semicircle}$$\textbf{(E)} \text{ a circle}$