14. Line $A B$

and $A C$ are
tangent to circle $O$ at
points $B$ and $C$,
respectively. $P$ is a
point on the circle,
and the distances from $P$ to $A B$ and $A C$
are 4 cm and 6 cm, respectively. The distance from $P$ to $B C$ is $\qquad$ cm.
$\begin{array}{l}\text { 14. As shown in the figure } P M \perp A B, \\ P N \perp A C, P Q \perp B C . P \text {, } \\ Q, C, N \text { are concyclic, } P \text {, } \\ Q, B, N \text { are concyclic, } \\ \angle M P Q \\ =180^{\circ}-\angle M B Q \\ =180^{\circ}-\angle N C Q=\angle N P Q, \\ \angle M Q P=\angle M B P=\angle B C P=\angle Q N P,\end{array}$
$\begin{array}{l}\therefore \triangle M P Q \sim \triangle Q P N . \text { Therefore, } \frac{M P}{P Q}=\frac{P Q}{N P}, \\ \text { hence, } P Q=\sqrt{M P \cdot N P}=2 \sqrt{6} \text { (cm). }\end{array}$