Olympiad Geometry Problem #30: Incenter, Equal Segments, Perpendicular

Here is an extremely beautiful problem that I found on the Art of Problem Solving Forum, but I don't know the original source. Just by drawing two equal segments, we get a nontrivial perpendicularity. Enjoy! Link below.

[vasilijeivanovic]

if you use the following theorem this problem becomes trivial:
the locus of points for which d(AB, P)+d(AC, P)+d(BC, P) is constant, is a line that is peripendicular to OI

[kerimbendov]

That is the exact solution that i came up with :D (Ps: it is not that hard)