AHSME 1960 Problem 15

Triangle $I$ is equilateral with side $A$, perimeter $P$, area $K$, and circumradius $R$ (radius of the circumscribed circle). Triangle $II$ is equilateral with side $a$, perimeter $p$, area $k$, and circumradius $r$. If $A$ is different from $a$, then: $\textbf{(A)}\ P:p = R:r \text{ } \text{only sometimes} \\ \textbf{(B)}\ P:p = R:r \text{ } \text{always} \\ \textbf{(C)}\ P:p = K:k \text{ } \text{only sometimes} \\ \textbf{(D)}\ R:r = K:k \text{ } \text{always} \\ \textbf{(E)}\ R:r = K:k \text{ } \text{only sometimes}$