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Making triangles - Iran NMO 1999 (Second Round) Problem2

Source:

October 4, 2010
symmetryvectorgeometry proposedgeometry

Problem Statement

ABCABC is a triangle with ∠B>45∘\angle{B}>45^{\circ} , ∠C>45∘\angle{C}>45^{\circ}. We draw the isosceles triangles CAM,BANCAM,BAN on the sides AC,ABAC,AB and outside the triangle, respectively, such that ∠CAM=∠BAN=90∘\angle{CAM}=\angle{BAN}=90^{\circ}. And we draw isosceles triangle BPCBPC on the side BCBC and inside the triangle such that ∠BPC=90∘\angle{BPC}=90^{\circ}. Prove that ΔMPN\Delta{MPN} is an isosceles triangle, too, and ∠MPN=90∘\angle{MPN}=90^{\circ}.
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