X₂₀–X₃₀

Euler Line Centers

Classical centers lying on the Euler line X₂ X₃, including the De Longchamps point, Schiffler point, Exeter point, far-out point, homothetic center of the orthic and tangential triangles, and the Euler infinity point.

5, 6, 7 3, 4, 5 Kepler Kimberling

X₂₀–X₃₀ (Euler Line Centers) — 10 representative centers (from 11 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — Centers 20-30 (Points on the Euler line)

2- 5, 140, 186, 199, 235, 237, 297, 376- 379,381- 384,

401- 475, 546- 550, 631, 632 (and others) lie on the Euler line.

X₂₀DE LONGCHAMPS POINT

X₂₀ = anticomplement of X₄

X₂₁SCHIFFLER POINT

Write I for the incenter; the Euler lines of the four triangles IBC, ICA, IAB, and ABC concur in X₂₁. This configuration extends to Kirikami-Schiffler points and generalizations found by Peter Moses, as introduced just before X₃₆₄₈.

X₂₂EXETER POINT

X₂₂ is the perspector of the circummedial triangle and the tangential triangle; also X₂₂ = X₅₅-of-the-tangential triangle if ABC is acute. See the note just before X₁₆₀₁ for a generalization.

X₂₃FAR-OUT POINT

X₂₃ = inverse-in-circumcircle of X₂

X₂₄PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE

X₂₄ is the perspector of ABC and orthic-of-orthic triangle.

X₂₅HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES

Constructed as indicated by the name; also X₂₅ is the pole of the orthic axis (the line having trilinear coefficients cos A : cos B : cos C) with respect to the circumcircle.

X₂₆CIRCUMCENTER OF THE TANGENTIAL TRIANGLE

Theorems involving X₂₆, published in 1889 by A. Gob, are discussed in

Roger A. Johnson, Advanced Euclidean Geometry, Dover, 1960, 259-260.

X₂₇CEVAPOINT OF ORTHOCENTER AND CLAWSON CENTER

X₂₇ is the cevapoint of orthocenter and Clawson center.

X₂₈CEVAPOINT OF X₁₉ AND X₂₅

X₂₈ is the cevapoint of X₁₉ and X₂₅.

X₂₉CEVAPOINT OF INCENTER AND ORTHOCENTER

X₂₉ is the cevapoint of incenter and orthocenter.

X₃₀EULER INFINITY POINT

X₃₀ is the point of intersection of the Euler line and the line at infinity. Thus, each of the lines listed below is parallel to the Euler line.