In triangle $ABC,$ let the angle bisectors be $\overline{BY}$ and $\overline{CZ}$. Given $AB = 12$, $AY = 12$, and $CY = 6$, find $BZ$.
A
12
12 Z Y
6
B C
Since BY is a bisector
AB / AY = BC / CY
12 / 12 = BC / 6
BC = 6
And since CZ is a bisector let BZ = x and AZ = 12 - x
BZ / BC = AZ / AY
x / 6 = (12 - x) / 12
12 x = 6 (12 - x)
12x = 72 - 6x
18x = 72
x = 72/ 18 = 4 = AZ
+357 
