$17 \cdot 59$ In an isosceles triangle, the angle bisector of one base angle divides the perimeter into two parts, 168 and 112. Then the length of the leg of the triangle is
(A) 80.
(B) 105 or 80.
(C) 105.
(D) None of the above.
(Shenyang, Liaoning Province, China Junior High School Mathematics Invitational Competition, 1986)
[Solution] Let $BE$ be the angle bisector of the base angle $\angle B$ in isosceles $\triangle ABC$. According to the angle bisector property and the given conditions,

or
$$
\frac{BC}{AB}=\frac{CE}{EA}=\frac{BC+CE}{AB+EA}=\frac{112}{168},
$$

or $\frac{BC}{AB}=\frac{CE}{EA}=\frac{BC+CE}{AB+EA}=\frac{168}{112}$.
Thus, $BC=\frac{112}{168} AB$ or $BC=\frac{168}{112} AB$,
and since $112+168=2AB+BC$, substituting the above equations yields
$$
AB=105 \text{ or } 80 \text{. }
$$

Therefore, the answer is $(B)$.