The Euler points are the midpoints , , of the segments which join the vertices , , and of a triangle and the orthocenter . They are three of the nine prominent points of a triangle through which the nine-point circle passes. The Euler points determine the Euler triangle .
Given a triangle , construct the orthic triangle . Then the Euler lines of the three corner triangles , and pass through the Euler points, and concur at a point on the nine-point circle of triangle such that one of
(1)
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(2)
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(3)
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holds (Thébault 1947, 1949; Thébault et al. 1951).