MEMO 2014 ekipno problem 5

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Dodao/la: arhiva
24. rujna 2014.
Let ABC be a triangle with AB < AC. Its incircle with centre I touches the sides BC, CA, and AB in the points D, E, and F respectively. The angle bisector AI intersects the lines DE and DF in the points X and Y respectively. Let Z be the foot of the altitude through A with respect to BC.

Prove that D is the incentre of the triangle XYZ.
Izvor: Srednjoeuropska matematička olimpijada 2014, ekipno natjecanje, problem 5