1973 AHSME Problem 35

In the unit circle shown in the figure, chords PQ and MN are parallel to the unit radius OR of the circle with center at O. Chords MP, PQ, and NR are each s units long and chord MN is d units long.

Of the three equations

\textbf{I.}\ d-s=1, \qquad \textbf{II.}\ ds=1, \qquad \textbf{III.}\ d^2-s^2=\sqrt{5}

those which are necessarily true are

\textbf{(A)}\ \textbf{I}\ \text{only} \qquad\textbf{(B)}\ \textbf{II}\ \text{only} \qquad\textbf{(C)}\ \textbf{III}\ \text{only} \qquad\textbf{(D)}\ \textbf{I}\ \text{and}\ \textbf{II}\ \text{only} \qquad\textbf{(E)}\ \textbf{I, II}\ \text{and} \textbf{ III}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.