MathDB
VMO 2018 P7

Source: Vietnam MO 2nd day 3rd problem (last problem)

January 12, 2018
geometry

Problem Statement

Acute scalene triangle ABCABC has GG as its centroid and OO as its circumcenter. Let Ha, Hb, HcH_a,\, H_b,\, H_c be the projections of A, B, CA,\, B,\, C on respective opposite sides and D, E, FD,\, E,\, F be the midpoints of BC, CA, ABBC,\, CA,\, AB in that order. GHa→, GHb→, GHc→\overrightarrow{GH_a},\, \overrightarrow{GH_b},\, \overrightarrow{GH_c} intersect (O)(O) at X, Y, ZX,\,Y,\,Z respectively. a. Prove that the circle (XCE)(XCE) pass through the midpoint of BHaBH_a b. Let M, N, PM,\, N,\, P be the midpoints of AX, BY, CZAX,\, BY,\, CZ respectively. Prove that DM↔, EN↔, FP↔\overleftrightarrow{DM},\, \overleftrightarrow{EN},\,\overleftrightarrow{FP} are concurrent.
Back to ProblemsView on AoPS