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A Concurrency from the Midpoints of Line Segments through the Circumcenter

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Let ABC be a triangle with circumcenter O. Let A', B', and C' be the perpendicular projections of A, B, and C onto BC, CA, and AB, respectively. Let
D=AO⋂BC
,
E=BO⋂AC
, and
F=CO⋂AB
. Let A'', B'', and C'' be the midpoints of AD, BE, and CF, respectively. Then A'A'', B'B'', and C'C'' are concurrent.
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