16.13 On a straight line, take five points $O, A, B, C, D$ in sequence, and let $O A=a$, $O B=b$, $O C=c$, $O D=d$, and $P$ be a point on the line between $B$ and $C$, such that $A P: P D = B P: P C$. Then $O P$ equals
(A) $\frac{b^{2}-b c}{a-b+c-d}$.
(B) $\frac{a c-b d}{a-b+c-d}$.
(C) $-\frac{b d+a c}{a-b+c-d}$.
(D) $\frac{b c+a d}{a+b+c+d}$.
(E) $\frac{a c-b d}{a+b+c+d}$.
(17th American High School Mathematics Examination, 1966)
[Solution] Let $O P=p$, then
$$
A P=p-a, \quad P D=d-p, \quad B P=p-b, \quad P C=c-p .
$$

Then $A P: P D=B P: P C$ can be expressed as
$$
\frac{(p-a):(d-p)=(p-b):(c-p),}{O_{B}}
$$

Simplifying, we get $[(a+c)-(b+d)] p=a c-b d$.
$$
\therefore
$$

Thus, the answer is $(B)$.
$$
p=\frac{a c-b d}{a-b+c-d} \text {. }
$$