Euler Points


The Euler points are the midpoints E_A, E_B, E_C of the segments which join the vertices A, B, and C of a triangle DeltaABC and the orthocenter H. 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 DeltaE_AE_BE_C.


Given a triangle DeltaABC, construct the orthic triangle DeltaH_AH_BH_C. Then the Euler lines of the three corner triangles DeltaAH_BH_C, DeltaBH_CH_A and DeltaCH_AH_B pass through the Euler points, and concur at a point P on the nine-point circle of triangle DeltaABC such that one of


holds (Thébault 1947, 1949; Thébault et al. 1951).

See also

Euler Triangle, Feuerbach's Theorem, Nine-Point Circle, Orthic Triangle

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Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., p. 6, 1995.Thébault, V. "Concerning the Euler Line of a Triangle." Amer. Math. Monthly 54, 447-453, 1947.Thébault, V. "Problem 4328." Amer. Math. Monthly 56, 39-40, 1949.Thébault, V.; Ramler, O. J.; and Goormaghtigh, R. "Solution to Problem 4328: Euler Lines." Amer. Math. Monthly 58, 45, 1951.

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Euler Points

Cite this as:

Weisstein, Eric W. "Euler Points." From MathWorld--A Wolfram Web Resource.

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