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Midpoint of foot of perpendicular and circumcenter

Source: Cyprus 2022 Junior TST-2 Problem 3

February 21, 2022
geometryisoscelescircumcircle

Problem Statement

Let ABCABC be an acute-angled triangle, and let D,ED, E and KK be the midpoints of its sides AB,ACAB, AC and BCBC respectively. Let OO be the circumcentre of triangle ABCABC, and let MM be the foot of the perpendicular from AA on the line BCBC. From the midpoint PP of OMOM we draw a line parallel to AMAM, which meets the lines DEDE and OAOA at the points TT and ZZ respectively. Prove that:
(a) the triangle DZEDZE is isosceles (b) the area of the triangle DZEDZE is given by the formula EDZE=BC⋅OK8E_{DZE}=\frac{BC\cdot OK}{8}
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