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three lines through midpoints of arcs

Source: IMO Shortlist 1997, Q25

September 25, 2004
geometrycircumcircleincenterparallelogramconcurrencyIMO Shortlist

Problem Statement

Let X,Y,Z X,Y,Z be the midpoints of the small arcs BC,CA,AB BC,CA,AB respectively (arcs of the circumcircle of ABC ABC). M M is an arbitrary point on BC BC, and the parallels through M M to the internal bisectors of ∠B,∠C \angle B,\angle C cut the external bisectors of ∠C,∠B \angle C,\angle B in N,P N,P respectively. Show that XM,YN,ZP XM,YN,ZP concur.
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