AHSME 1960 Problem 35

From point $P$ outside a circle, with a circumference of $10$ units, a tangent is drawn. Also from $P$ a secant is drawn dividing the circle into unequal arcs with lengths $m$ and $n$. It is found that $t_1$, the length of the tangent, is the mean proportional between $m$ and $n$. If $m$ and $t$ are integers, then $t$ may have the following number of values: $\textbf{(A)}\ \text{zero} \textbf{(B)}\ \text{one} \textbf{(C)}\ \text{two} \textbf{(D)}\ \text{three} \textbf{(E)}\ \text{infinitely many}$