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Concurrents and perpendicular bisector

Source: Greek MO 2012 - P3

May 12, 2013
geometryparallelogramangle bisectorperpendicular bisectorgeometry proposed

Problem Statement

Let an acute-angled triangle ABCABC with AB<AC<BCAB<AC<BC, inscribed in circle c(O,R)c(O,R). The angle bisector ADAD meets c(O,R)c(O,R) at KK. The circle c1(O1,R1)c_1(O_1,R_1)(which passes from A,DA,D and has its center O1O_1 on OAOA) meets ABAB at EE and ACAC at ZZ. If M,NM,N are the midpoints of ZCZC and BEBE respectively, prove that: a)the lines ZE,DM,KCZE,DM,KC are concurrent at one point TT. b)the lines ZE,DN,KBZE,DN,KB are concurrent at one point XX. c)OKOK is the perpendicular bisector of TXTX.
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