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Part of 2008 Greece JBMO TST

Problems(1)

circumcenter of a triangle lies on a given circle

Source: Greece JBMO TST 2008 p1

4/29/2019
Given a point AA that lies on circle c(o,R)c(o,R) (with center OO and radius RR). Let (e)(e) be the tangent of the circle cc at point AA and a line (d)(d) that passes through point OO and intersects (e)(e) at point MM and the circle at points B,CB,C (let BB lie between OO and AA). If AM=R3AM = R\sqrt3 , prove that a) triangle AMCAMC is isosceles. b) circumcenter of triangle AMCAMC lies on circle cc .
geometryCircumcentertangentisoscelescircle