MathDB
Concyclic

Source: Chinese National Olympiad 2009 P1

January 10, 2009
trigonometrygeometrytrapezoidGausscyclic quadrilateralcongruent trianglesangle bisector

Problem Statement

Given an acute triangle PBC PBC with PB≠PC. PB\neq PC. Points A,D A,D lie on PB,PC, PB,PC, respectively. AC AC intersects BD BD at point O. O. Let E,F E,F be the feet of perpendiculars from O O to AB,CD, AB,CD, respectively. Denote by M,N M,N the midpoints of BC,AD. BC,AD. (1) (1): If four points A,B,C,D A,B,C,D lie on one circle, then EM\cdot FN \equal{} EN\cdot FM. (2) (2): Determine whether the converse of (1) (1) is true or not, justify your answer.
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