The Greatest of the Rejected Ones

$PB=3, PC=11$ $PB+PC=14$ By definition of an isosceles triangle, $A$ must be located above the midpoint of $BC$ ; if $D$ were the midpoint, $BD$ would be of length $\frac{PB}{2}=\frac{14}{2}=7$ . Remember that $A$ is above the midpoint. Because a line segment with 3 points is not a triangle, we can use the lengths of $BD$ and $PB$ to find our answer: $BD-PB=7-3=4$ Final answer: $PA>4$

Bonus: $PA\gt abs\left(\frac{BC}{2}-PB\right)$