Let
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
be a point inside a triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
such that
Let
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
,
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
be the incenters of triangles
![APB](/media/m/4/e/4/4e497177a6f8b18739694beb5d551a48.png)
,
![APC](/media/m/6/6/2/6628b42a216b91f6ec57f035b8f7c790.png)
, respectively. Show that the lines
![AP](/media/m/7/b/0/7b05fe3b464ec24a15fa5701f4d14b61.png)
,
![BD](/media/m/1/1/f/11f65a804e5c922ee28a53b1df04d138.png)
,
![CE](/media/m/3/3/7/33771853199c7fd8dc7faa2d4a37425d.png)
meet at a point.
%V0
Let $P$ be a point inside a triangle $ABC$ such that
$$\angle APB - \angle ACB = \angle APC - \angle ABC.$$
Let $D$, $E$ be the incenters of triangles $APB$, $APC$, respectively. Show that the lines $AP$, $BD$, $CE$ meet at a point.