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Part of 1995 Turkey MO (2nd round)

Problems(1)

Turkish MO 1995 P2

Source: Turkish Mathematical Olympiad 2nd Round 1995

9/30/2006
Let ABCABC be an acute triangle and let k1,k2,k3k_{1},k_{2},k_{3} be the circles with diameters BC,CA,ABBC,CA,AB, respectively. Let KK be the radical center of these circles. Segments AK,CK,BKAK,CK,BK meet k1,k2,k3k_{1},k_{2},k_{3} again at D,E,FD,E,F, respectively. If the areas of triangles ABC,DBC,ECA,FABABC,DBC,ECA,FAB are u,x,y,zu,x,y,z, respectively, prove that u2=x2+y2+z2.u^{2}=x^{2}+y^{2}+z^{2}.
geometrytrigonometrygeometric transformationreflectioncircumcircleradical axisgeometry unsolved