2006 IMO Problems/Problem 1
Let be triangle with incenter . A point in the interior of the triangle satisfies . Show that , and that equality holds if and only if
We have (1) and similarly (2). Since , we have (3).
By (1), (2), and (3), we get ; hence are concyclic.
Let ray meet the circumcircle of at point . Then, by the Incenter-Excenter Lemma, .
Finally, (since triangle APJ can be degenerate, which happens only when ), but ; hence and we are done.
By Mengsay LOEM , Cambodia IMO Team 2015
latexed by tluo5458 :)