Unformatted text preview: that the midpoint of segment PQ is the incenter of 4 ABC . 6. (IMO 1998) A convex quadrilateral ABCD has perpendicular diagonals. The perpendicular bisectors of AB and CD meet at a unique point P inside ABCD . Prove that ABCD is cyclic if and only if triangles ABP and CDP have equal areas. 7. (IMO 2001) Let ABC be an acute triangle with circumcenter O and with ∠ C ≥ ∠ B + 30 ◦ . Let P be the foot of the altitude from A to BC . Prove that ∠ A + ∠ COP < 90 c irc . 8. (IMO 2009) Let ABC be a triangle with circumcenter O . The points P and Q are interior points of sides AC and AB , respectively. Let K , L , and M be midpoints of the segments BP , CQ , and PQ , respectively, and let Σ be the circle passing through K , L , and M . Suppose that the line PQ is tangent to the circle Σ Prove that OP = OQ ....
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 Spring '09
 B
 Geometry, Angles, Integers, Law Of Cosines, triangle, Ninepoint circle

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