1975 AHSME Problems/Problem 11

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Problem

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

$\textbf{(A)} \text{ a circle with one point deleted} \qquad \\ \textbf{(B)} \text{ a circle if the distance from } P \text{ to the center of } K \text{ is less than one half the radius of } K; \\ \text{otherwise a circular arc of less than } 360^{\circ} \qquad \\ \textbf{(C)} \text{ a semicircle with one point deleted} \qquad \\ \textbf{(D)} \text{ a semicircle} \qquad  \textbf{(E)} \text{ a circle}$

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