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BC^2/ MZ^2 >= 8 R U_a / (MD) (ME)

Source: 1988 Greece MO Grade XI p2

September 7, 2024
geometrygeometric inequality

Problem Statement

Let ABCABC be a triangle inscribed in circle C(O,R)C(O,R). Let MM ba apoint on the arc BCBC . Let D,E,ZD,E,Z be the feet of the perpendiculars drawn from MM on lines AB,AC,BCAB,AC,BC respectively. Prove that (BC)2(MZ)2≥8RUa(MD)⋅(ME)\frac{(BC)^2}{(MZ)^2} \ge 8\frac{R U_a}{(MD)\cdot(ME)} where UaU_a is the altitude drawn on BCBC.
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