Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be triangle with incenter
![I](/media/m/3/8/6/38689d6affa9ba35368ca4d3d76ea147.png)
. A point
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
in the interior of the triangle satisfies
![\angle PBA+\angle PCA = \angle PBC+\angle PCB.](/media/m/9/f/9/9f9e800bab1cf586107fb306ac235851.png)
Show that
![AP \geq AI](/media/m/f/5/8/f589f13dc4c54a617e4d15ddf051f964.png)
, and that equality holds if and only if
![P=I](/media/m/d/6/b/d6bbcffbc7b1a06a21a809c03d9e1ab4.png)
.
%V0
Let $ABC$ be triangle with incenter $I$. 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 $P=I$.