IMO - SL-2008 - G4

In an acute triangle ABC segments BE and CF are altitudes. Two circles passing through the point A anf F and tangent to the line BC at the points P and Q so that B lies between C and Q. Prove that lines PE and QF intersect on the circumcircle of triangle AEF.

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IMO2005-P5

Let ABCD be a xed convex quadrilateral with BC = DA and BC not parallel with DA. Let two variable points E and F lie of the sides BC and DA, respectively and satisfy BE = DF. The lines AC and BDmeet at P, the lines BD and EF meet at Q, the lines EF and AC meet at R. Prove that the circumcircles of the triangles PQR, as E and F vary, have a common point other than P.

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IMO 2005-P1

Six points are chosen on the sides of an equilateral triangle ABC: A1, A2 on BC, B1, B2 on CA and C1, C2 on AB, such that they are the vertices of a convex hexagon A1A2B1B2C1C2 with equal side lengths. Prove that the lines A1B2;B1C2 and C1A2 are concurrent.

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