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A Concurrency Involving the Midpoints of the Sides and the Altitudes

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Let ABC be a triangle and let A', B', and C' be the midpoints of the sides opposite A, B, and C. Let A", B'', and C'' be the midpoints of the altitudes from A, B, and C, respectively. Then A'A'', B'B'', and C'C'' are concurrent.
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