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if l contains circumcenter of ABC, then l' contains it's Euler circle center

Source: 2010 Balkan Shortlist G4 BMO

April 4, 2020
CircumcenterEuler Circlegeometrymidpointprojectionscollinear

Problem Statement

Let ABCABC be a given triangle and ℓ\ell be a line that meets the lines BC,CABC, CA and ABAB in A1,B1A_1,B_1 and C1C_1 respectively. Let A′A' be the midpoint, of the segment connecting the projections of A1A_1 onto the lines ABAB and ACAC. Construct, analogously the points B′B' and C′C'. (a) Show that the points A′,B′A', B' and C′C' are collinear on some line ℓ′\ell'. (b) Show that if ℓ\ell contains the circumcenter of the triangle ABCABC, then ℓ′\ell' contains the center of it's Euler circle.
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