12. In the triangle $A B C, A B=14, B C=16, A C=26, M$ is the midpoint of $B C$ and $D$ is the point on $B C$ such that $A D$ bisects $\angle B A C$. Let $P$ be the foot of the perpendicular from $B$ onto $A D$. Determine the length of $P M$.
12. Ans: 6

Extend $B P$ meeting $A C$ at $E$. Then $A B E$ is an isosceles triangle with $A B=A E$ and $B P=P E$. As $P$ and $M$ are the midpoints of $B E$ and $B C$ respectively, we have $P M$ is parallel to $E C$ and $P M=E C / 2=(26-14) / 2=6$.