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
and similarly Since , we have
It follows that Hence, and 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 :)
minor edits by lpieleanu