Πέμπτη, 12 Οκτωβρίου 2017

2003 JBMO problem 3 (BUL)

proposed by Ch. Lozanov, Bulgaria

Let ${D, E, F}$  be the midpoints of the arcs ${BC, CA, AB}$  on the circumcircle of a triangle ${ABC}$  not containing the points ${A, B, C}$, respectively. Let the line ${DE}$  meets ${BC}$  and ${CA}$  at ${G}$  and ${H}$, and let ${M}$  be the midpoint of the segment ${GH}$. Let the line ${FD}$  meet ${BC}$ and ${AB}$  at ${K}$  and ${J}$, and let ${N}$  be the midpoint of the segment ${KJ}$.
a) Find the angles of triangle ${DMN}$,

b) Prove that if ${P}$  is the point of intersection of the lines ${AD}$  and ${EF}$, then the circumcenter of triangle ${DMN}$  lies on the circumcircle of triangle ${PMN}$.


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