Question 1
Let $ABC$ be a triangle with incentre $I$. A point $P$ in the interior of the triangle satisfies
$$
\angle PBA + \angle PCA = \angle PBC + \angle PCB.
$$
Show that $AP \geq AI$ and that equality holds if and only if $P$ coincides with $I$.
$$
\angle PBA + \angle PCA = \angle PBC + \angle PCB.
$$
Show that $AP \geq AI$ and that equality holds if and only if $P$ coincides with $I$.