IMO Shortlist 2006 problem G4

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Let ABC be a triangle such that \widehat{ACB} < \widehat{BAC} < \frac {\pi}{2}. Let D be a point of [AC] such that BD = BA. The incircle of ABC touches [AB] at K and [AC] at L. Let J be the center of the incircle of BCD. Prove that (KL) intersects [AJ] at its middle.
Izvor: Međunarodna matematička olimpijada, shortlist 2006