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Geometry

Source: 2016 China Second Round Olympiad Problem 2

January 8, 2017
geometry

Problem Statement

Let X,YX,Y be two points which lies on the line BCBC of △ABC(X,B,C,Ylies in sequence)\triangle ABC(X,B,C,Y\text{lies in sequence}) such that BX⋅AC=CY⋅ABBX\cdot AC=CY\cdot AB, O1,O2O_1,O_2 are the circumcenters of △ACX,△ABY\triangle ACX,\triangle ABY, O1O2∩AB=U,O1O2∩AC=VO_1O_2\cap AB=U,O_1O_2\cap AC=V. Prove that △AUV\triangle AUV is a isosceles triangle.
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