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concurrent wanted, midpoints , arc midpoints of circumcircle related

Source: Switzerland 2020 Swiss MO p2

December 8, 2020
geometryconcurrencyconcurrentarc midpoint

Problem Statement

Let ABCABC be an acute triangle. Let MA,MBM_A, M_B and MCM_C be the midpoints of sides BC,CABC,CA, respectively ABAB. Let MA′,MB′M'_A , M'_B and MC′M'_C be the the midpoints of the arcs BC,CABC, CA and ABAB respectively of the circumscriberd circle of triangle ABCABC. Let PAP_A be the intersection of the straight line MBMCM_BM_C and the perpendicular to MB′MC′M'_BM'_C through AA. Define PBP_B and PCP_C similarly. Show that the straight line MAPA,MBPBM_AP_A, M_BP_B and MCPCM_CP_C intersect at one point.
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