Triangle area probability

Let $\angle BAD = \angle DAC$ where $D \in \overline{BC}$ .

Notice that the angle bisctor is the locus of points $P$ such that it's equidistant from $\overline{AB}$ and $\overline{AC}$ .

Thus we have to compute $S \Delta BAD$ because the points on $\Delta BAD$ are nearer or equal compared to $\overline{AC}$ .

Since $\Delta BAD$ and $\Delta ABC$ have the same height, we get that $\dfrac{S \Delta BAD}{S \Delta ABC}$ $=$ $\dfrac{\overline{BD}}{\overline{DC}} = \dfrac{5}{7}$ .

So the answer is $\boxed{\dfrac{5}{7}}$ .

Remarks: This is part of the Philippine Mathematical Olympiad, Qualifying Stage.