In triangle $ABC,$ let the angle bisectors be $\overline{BY}$ and $\overline{CZ}$. Given $AB = 14$, $AY = 14$, and $CY = 8$, find $BZ$.
A
14
14 Z Y
8
B C
Since BY is an angle bisector
AB / AY = BC / CY
14 / 14 = BC / 6
BC = 8
Call BZ = x and AZ = 14 - x
since CZ is an angle bisector
BZ / BC = AZ / AY
x / 8 = (14 - x) / 14
14x = 8 ( 14 - x)
14x = 112 - 8x
20x = 112
x = 112 / 20 = 28 / 5 = BZ
+1443 
