17・187 Given $\triangle A B C$ is determined by the given side $a$ (opposite angle $A$), angle $B$, and the altitude $h_{c}$ from $C$, $N$ is the number of non-overlapping triangles thus determined. Then $N$ is equal to
(A) 1.
(B) 2.
(C) 0.
(D) Infinite.
(E) 0 or Infinite.
(18th American High School Mathematics Examination, 1967)
[Solution] As shown in the figure, let $BC = a$, and draw a line $l$ through $B$ such that the angle between $l$ and $BC$ is $\angle B$.
(1) If the distance from $C$ to $l$ is $h_c$, then vertex $A$ can be located at any point on $l$ (except point $B$), meaning there are infinitely many solutions;
(2) If the distance from $C$ to $l$ is not equal to $h_c$, then the problem has no solution.
In conclusion, the answer is $(E)$.