AHSME 1960 Problem 32

In this figure the center of the circle is $O$. $AB \perp BC$, $ADOE$ is a straight line, $AP = AD$, and $AB$ has a length twice the radius. Then:

$\textbf{(A)} AP^2 = PB \times AB \\ \textbf{(B)}\ AP \times DO = PB \times AD \\ \textbf{(C)}\ AB^2 = AD \times DE \\ \textbf{(D)}\ AB \times AD = OB \times AO \\ \textbf{(E)}\ \text{none of these}$