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Show that the line AX is perpendicular to BC

Source: Middle European Mathematical Olympiad 2011 - Team Compt. T-6

September 6, 2011
geometrycircumcircletrigonometryPythagorean Theoremtrig identitiesLaw of Cosines

Problem Statement

Let ABCABC be an acute triangle. Denote by B0B_0 and C0C_0 the feet of the altitudes from vertices BB and CC, respectively. Let XX be a point inside the triangle ABCABC such that the line BXBX is tangent to the circumcircle of the triangle AXC0AXC_0 and the line CXCX is tangent to the circumcircle of the triangle AXB0AXB_0. Show that the line AXAX is perpendicular to BCBC.
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