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Euler Triangle


EulerTriangle

The Euler triangle of a triangle DeltaABC is the triangle DeltaE_AE_BE_C whose vertices are the midpoints of the segments joining the orthocenter H with the respective vertices. The vertices of the triangle are known as the Euler points, and lie on the nine-point circle. The Euler triangle is congruent and homothetic to the medial triangle and perspective to the orthic triangle (Kimberling 1998, p. 158).

It has trilinear vertex matrix

 [2x+y+z sinAsecB sinAsecC; sinBsecA x+2y+z sinBsecC; sinCsecA sinCsecB x+y+2z],

where x=tanA, y=tanB, and z=tanC.

The following table gives the centers of the Euler triangle in terms of the centers of the reference triangle for Kimberling centers X_n with n<=100.

A spherical triangle is sometimes also called Euler's triangle.


See also

Euler's Number Triangle, Euler Points, Nine-Point Circle, Second-Order Eulerian Triangle

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References

Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Referenced on Wolfram|Alpha

Euler Triangle

Cite this as:

Weisstein, Eric W. "Euler Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulerTriangle.html

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