MathDB

2

Part of 2011 Rioplatense Mathematical Olympiad, Level 3

Problems(1)

Triangle similarity

Source: Rioplatense Olympiad 2011, Level 3, Problem 2

8/29/2014
Let ABCABC an acute triangle and HH its orthocenter. Let EE and FF be the intersection of lines BHBH and CHCH with ACAC and ABAB respectively, and let DD be the intersection of lines EFEF and BCBC. Let Γ1\Gamma_1 be the circumcircle of AEFAEF, and Γ2\Gamma_2 the circumcircle of BHCBHC. The line ADAD intersects Γ1\Gamma_1 at point I≠AI \neq A. Let JJ be the feet of the internal bisector of ∠BHC\angle{BHC} and MM the midpoint of the arc BC⌢\stackrel{\frown}{BC} from Γ2\Gamma_2 that contains the point HH. The line MJMJ intersects Γ2\Gamma_2 at point N≠MN \neq M. Show that the triangles EIFEIF and CNBCNB are similar.
geometrycircumcirclepower of a pointradical axisgeometric transformationangle bisectorgeometry unsolved