In triangle ABC, let the angle bisectors be BY and CZ. Given AB = 16, AY = 16, and CY = 16, find BC and BZ.
+30 Note that since \(BY\) is both an angle bisector and a median, triangle \(ABC\) is isosceles with \(AB=BC\). Therefore, \(BC = \boxed{16}\). A simple application of the angle bisector theorem (using \(CZ\) as the angle bisector) yields \(\frac{BZ}{AZ} = \frac{BC}{AC} = \frac{1}{2},\) so \(BZ = \frac{1}{3} \cdot 16 = \boxed{ \frac{16}{3}}.\)
