X₃₅₁₅–X₃₆₀₁

Euler Line Intercepts & Named Geometric Centers

Euler line intercepts of central lines (X₃₅₁₅–X₃₅₅₂), together with the Dosa, Pappus, Moses circumcircle/infinity, Don Quixote, Beltrami-Euler, Beltrami-Parry, Myakishev conic, Dixit, Odehnal, and Liu points (X₃₅₅₅–X₃₆₀₁).

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

X₃₅₁₅–X₃₆₀₁ (Euler Line Intercepts & Named Geometric Centers) — 16 representative centers (from 87 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

X₃₅₁₅EULER LINE INTERCEPT OF LINE X₁₅₄X_185

X₃₅₁₅ = crosspoint of X₂₅₀ and X₁₃₀₁

X₃₅₁₆EULER LINE INTERCEPT OF LINE X₆₄X_184

X₃₅₁₆ = homothetic center of Trinh triangle and mid-triangle of orthic and circumorthic triangles

X₃₅₁₇EULER LINE INTERCEPT OF LINE X₁₅₄X_389

X₃₅₁₇ = homothetic center of tangential triangle and mid-triangle of orthic and circumorthic triangles

X₃₅₁₈EULER LINE INTERCEPT OF LINE X₅₁X_54

X₃₅₁₈ = X₁₁₇₉-Ceva conjugate of X₄

X₃₅₁₉ISOGONAL CONJUGATE OF X₃₅₁₈

X₃₅₁₉ is the isogonal conjugate of X₃₅₁₈.

X₃₅₂₀EULER LINE INTERCEPT OF LINE X₅₄X_74

X₃₅₂₀ = reflection of X₄ in X₁₅₉₄

X₃₅₂₁ISOGONAL CONJUGATE OF X₃₅₂₀

X₃₅₂₁ is the isogonal conjugate of X₃₅₂₀.

X₃₅₂₂EULER LINE INTERCEPT OF LINE X₈X_165

Other recent results on representations of points on the Euler line are given in

X₃₅₂₃EULER LINE INTERCEPT OF LINE X₃₅X_390

X₃₅₂₃ = anticomplement of X₃₀₉₀

X₃₅₂₄EULER LINE INTERCEPT OF LINE X₃₅X_1058

X₃₅₂₄ = trisector nearest X₃ of segment X₂X₃

X₃₅₂₅EULER LINE INTERCEPT OF LINE X₆₉X_575

X₃₅₂₅ = crosspoint of X₃₃₁₆ and X₃₃₁₇

X₃₅₂₆EULER LINE INTERCEPT OF LINE X₄₉X_182

X₃₅₂₆ = midpoint of X₃₀₉₀ and X₃₅₂₃

X₃₅₂₇ISOGONAL CONJUGATE OF X₆₃₁

X₃₅₂₇ is the isogonal conjugate of X₆₃₁.

X₃₅₂₈EULER LINE INTERCEPT OF LINE X₃₅X_1056

X₃₅₂₈ = pole of van Aubel line wrt conic {{X₁₃,X₁₄,X₁₇,X₁₈,X₂₀}}

X₃₅₂₉EULER LINE INTERCEPT OF LINE X₅₁₆X_944

X₃₅₂₉ = anticomplement of X₃₈₂

X₃₅₃₀EULER LINE INTERCEPT OF LINE X₁₅X_3411

X₃₅₃₀ = complement of X₅₄₆

X₃₅₃₁INTERSECTION OF LINES X(3,373) AND X(69,381)

X₃₅₃₁ = perspector of circle (X₃,2R)

X₃₅₃₂INTERSECTION OF LINES X(6,1204) AND X(64,1620)

Let DEF be the cevian triangle of X₆₉ and A' the center of the circle that passes through D and through points where the line EF cuts to the circumcircle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₃₅₃₂. (Angel Montesdeoca, October 30, 2019)

X₃₅₃₃EULER LINE INTERCEPT OF LINE X₃₇₁X_3317

X₃₅₃₃ = pole of Napoleon axis wrt conic {{X₂,X₁₅,X₁₆,X₁₇,X₁₈}}

X₃₅₃₄EULER LINE INTERCEPT OF LINE X₁₅₄X_2777

X₃₅₃₄ = Stammler isogonal conjugate of X₆

X₃₅₃₅EULER LINE INTERCEPT OF LINE X₂₇₅X_3317

X₃₅₃₅ = inverse-in-orthocentroidal-circle of X₃₅₃₆

X₃₅₃₆EULER LINE INTERCEPT OF LINE X₁₉X_587

X₃₅₃₆ = inverse-in-orthocentroidal-circle of X₃₅₃₅

X₃₅₃₇EULER LINE INTERCEPT OF LINE X₁₀₃₈X_1058

As a point on the Euler line, X₃₅₃₇ has Shinagawa coefficients \((2E, -2E + F)\).

X₃₅₃₈EULER LINE INTERCEPT OF LINE X₁₀₃₈X_1056

X₃₅₃₈ = {X₃₇₆,X₆₃₁}-harmonic conjugate of X₂₄

X₃₅₃₉EULER LINE INTERCEPT OF LINE X₁₀₅₆X_3084

X₃₅₃₉ = inverse-in-orthocentroidal-circle of X₃₅₄₀

X₃₅₄₀EULER LINE INTERCEPT OF LINE X₁₀₅₆X_3083

X₃₅₄₀ = inverse-in-orthocentroidal-circle of X₃₅₃₉

X₃₅₄₁EULER LINE INTERCEPT OF LINE X₃₃X_499

X₃₅₄₁ = inverse-in-orthocentroidal-circle of X₃₅₄₂

X₃₅₄₂EULER LINE INTERCEPT OF LINE X₃₃X_498

X₃₅₄₂ = inverse-in-orthocentroidal-circle of X₃₅₄₁

X₃₅₄₃EULER LINE INTERCEPT OF LINE X₁₄₆X_148

X₃₅₄₃ = midpoint of X₂ and X₃₁₄₆

X₃₅₄₄EULER LINE INTERCEPT OF LINE X₆X_14842

X₃₅₄₅EULER LINE INTERCEPT OF LINE X₁₁X_1056

X₃₅₄₅ = midpoint of X₄ and X₃₅₂₄

X₃₅₄₆EULER LINE INTERCEPT OF LINE X₄₉₈X_1038

X₃₅₄₆ = complement of X₃₀₈₉

X₃₅₄₇EULER LINE INTERCEPT OF LINE X₆₉X_155

As a point on the Euler line, X₃₅₄₇ has Shinagawa coefficients \((E + 2F, -E)\).

X₃₅₄₈EULER LINE INTERCEPT OF LINE X₆₈X_125

X₃₅₄₈ = complement of X₃₅₄₂

X₃₅₄₉EULER LINE INTERCEPT OF LINE X₆₈X_184

X₃₅₄₉ = complement of X₃₅₄₁

X₃₅₅₀INTERSECTION OF LINES X₁X_3 AND X₃₁X_43

X₃₅₅₀ = X₉₈₃-Ceva conjugate of X₁

X₃₅₅₁ISOGONAL CONJUGATE OF X₃₅₅₀

X₃₅₅₁ is the isogonal conjugate of X₃₅₅₀.

X₃₅₅₂EULER LINE INTERCEPT OF LINE X₃₂X_99

X₃₅₅₂ = cevapoint of X₃ and X₃₃₆₀

X₃₅₅₃INTERSECTION OF LINES X₁X_6 AND X₁₉X_41

X₃₅₅₃ = {X₁,X₆}-harmonic conjugate of X₃₅₅₄

X₃₅₅₄INTERSECTION OF LINES X₁X_6 AND X₁₉X_604

X₃₅₅₄ = crosspoint of X₁ and X₁₄₂₂

X₃₅₅₅DOSA POINT

X₃₅₅₅ = anticomplement of X₃₄₇₉₀

X₃₅₅₆INTERSECTION OF LINES X₂₅X_65 AND X₃₁X_56

X₃₅₅₆ = X₆₃-Ceva conjugate of X₆

X₃₅₅₇1st PAPPUS POINT

X₃₅₅₇ was submitted and called the Pappus point by Roland Bacher (March 11, 2009); coordinates and the related point X₃₅₅₈ were found by Peter Moses (March 12, 2009).

X₃₅₅₈2nd PAPPUS POINT

X₃₅₅₈ = crosssum of PU(119)

X₃₅₅₉SS(sin A → cos A) OF X₃₃₃

X₃₅₅₉ = X₁₈₉₆-Ceva conjugate of X₂₉

X₃₅₆₀SS(sin A → cos A) OF X₉₄₀

X₃₅₆₀ = complement of X₆₈₅₀

X₃₅₆₁SS(sin A → cos A) OF X₅₉₅

X₃₅₆₁ = crosssum of X₂₃₁₀ and X₂₅₀₁

X₃₅₆₂SS(sin A → cos A) OF X₁₆₂₁

X₃₅₆₂ = crosssum of X₆₄₇ and X₂₃₁₀

X₃₅₆₃1st MOSES CIRCUMCIRCLE POINT

X₃₅₆₃ = reflection of X₃₅₆₅ in X₃

X₃₅₆₄1st MOSES INFINITY POINT

X₃₅₆₄ = crossdifference of every pair of points on line X₆X₉₂₄

X₃₅₆₅2nd MOSES CIRCUMCIRCLE POINT

X₃₅₆₅ = reflection of X₃₅₆₃ in X₃

X₃₅₆₆2nd MOSES INFINITY POINT

X₃₅₆₆ = crossdifference of every pair of points on line X₆X₁₁₉₆

X₃₅₆₇DON QUIXOTE POINT

The other two Spanish Points are the Sancho Panza Point X₃₀₆₀ and the Miguel de Cervantes Point X₁₄₃.

X₃₅₆₈BELTRAMI-EULER POINT

Let P(2) and U(2) be the 1st and 2nd Beltrami points (as indexed at Bicentric Pairs, accessible using the Tables button at the top of ETC), and let P(40) and U(40) be the isogonal conjugates of P(2) and U(2), respectively. Then X₃₅₆₈ is the point of intersection of the lines P(2)P(40) and U(2)U(40). The name “Beltrami-Euler Point” signifies the fact that X₃₅₆₈ lies on the Euler line. See also X₂₉₆₆ and X₃₅₆₉. Contributed by Chris van Tienhoven, June 7, 2009.

X₃₅₆₉BELTRAMI-PARRY POINT

X₃₅₆₉ = perspector of hyperbola {{A, B, C,X₆,X₂₃₂}}

X₃₅₇₀INTERSECTION OF LINES P(6)U(8) AND P(8)U(6)

The points given by trilinears P(6) =\(b : c : a\) and U(6) =\(c : a : b\) are indexed at Bicentric Pairs (accessible using the Tables button at the top of ETC), and likewise for their isogonal conjugates P(8) =\(1/b : 1/c : 1/a\) and U(8) =\(1/c : 1/a : 1/b\). The bicentric pair of lines P(6)U(8) and P(8)U(6) concur in X₃₅₇₀. Contributed by Peter Moses, July 7, 2009..

X₃₅₇₁INTERSECTION OF LINES P(6)P(8) AND U(6)U(8)

The points given by trilinears P(6) =\(b : c : a\) and U(6) =\(c : a : b\) are indexed at Bicentric Pairs (accessible using the Tables button at the top of ETC), and likewise for their isogonal conjugates P(8) =\(1/b : 1/c : 1/a\) and U(8) =\(1/c : 1/a : 1/b\). The bicentric pair of lines P(6)P(8) and U(6)U(8) concur in X₃₅₇₁. Contributed by Peter Moses, July 7, 2009.

X₃₅₇₂INTERSECTION OF LINES P(6)U(6) AND P(8)U(8)

The points given by trilinears P(6) =\(b : c : a\) and U(6) =\(c : a : b\) are indexed at Bicentric Pairs (accessible using the Tables button at the top of ETC), and likewise for their isogonal conjugates P(8) =\(1/b : 1/c : 1/a\) and U(8) =\(1/c : 1/a : 1/b\). The pair of central lines P(6)U(6) and P(8)U(8) concur in X₃₅₇₂. Contributed by Peter Moses, July 7, 2009.

X₃₅₇₃TRILINEAR PRODUCT X₆*X₃₅₇₀

X₃₅₇₃ = cevapoint of X₂₃₈ and X₆₅₉

X₃₅₇₄INTERSECTION OF LINES X₄X_54 AND X₅X_51

X₃₅₇₄ = midpoint of X₄ and X₅₄

X₃₅₇₅EULER LINE INTERCEPT OF THE LINE X₆₄X_66

X₃₅₇₅ = reflection of X₁₈₈₅ in X₄

X₃₅₇₆INTERSECTION OF LINES X₁X_3 AND X₂X_515

Let P_a be the parabola with focus A and directrix BC, and define P_b and P_c cyclically. X₃₅₇₆ is the centroid of the six points of tangency of lines from X₁ to P_a, P_b, and P_c. (Randy Hutson, December 2, 2017)

X₃₅₇₇ISOGONAL CONJUGATE OF X₃₅₇₆

X₃₅₇₇ is the isogonal conjugate of X₃₅₇₆.

X₃₅₇₈INTERSECTION OF LINES X₂X_6 AND X₈X_30

X₃₅₇₈ = X₇₅-Ceva conjugate of X₁₁₂₅

X₃₅₇₉INTERSECTION OF LINES X₁X_3 AND X₁₀X_30

X₃₅₇₉ = X₃₂₁₀-Ceva conjugate of X₃₇

X₃₅₈₀INTERSECTION OF LINES X₂X_6 AND X₃₀X_74

X₃₅₈₀ = complement of X₃₂₃

X₃₅₈₁INTERSECTION OF LINES X₃X_6 AND X₃₀X_74

Let A₁B₁C₁ and A₂B₂C₂ be the 1st and 2nd Ehrmann circumscribing triangles. Let A* be the crosssum of A₁ and A₂, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X₃₅₈₁.

X₃₅₈₂INTERSECTION OF LINES X₁X_2 AND X₁₁X_30

X₃₅₈₃INTERSECTION OF LINES X₁X_4 AND X₁₁X_30

X₃₅₈₃ = crosspoint of X₇₉ and X₈₀

X₃₅₈₄INTERSECTION OF LINES X₁X_2 AND X₁₂X_30

X₃₅₈₅INTERSECTION OF LINES X₁X_4 AND X₁₂X_30

X₃₅₈₅ = reflection of X₃₅ in X₁₂

X₃₅₈₆INTERSECTION OF LINES X₁X_4 AND X₃₀X_57

X₃₅₈₆ = insimilicenter of hexyl and 2nd Johnson-Yff circles; the exsimilicenter is X₉₆₁₄

X₃₅₈₇INTERSECTION OF LINES X₁X_3 AND X₉X_30

X₃₅₈₈CENTER OF THE MYAKISHEV CONIC

X₃₅₈₈ = X₈X₅₇₃∩X(2269,3057), where

X₃₅₈₉ISOGONAL CONJUGATE OF X₃₁₀₈

P' = 2u + v + w : u + 2v + w : u + v +2w.

X₃₅₉₀1st DIXIT POINT

The radius of circle T_A is |BC|/6, and the Kiepert angle (e.g., B-to-C-to-X₃₈₉₀) is arctan(2/3).

X₃₅₉₁2nd DIXIT POINT

This point is obtained in the manner of X₃₅₉₀ using the inward semicircles instead of outward.

X₃₅₉₂ISOGONAL CONJUGATE OF 1st DIXIT POINT

X₃₅₉₂ is the isogonal conjugate of 1st dixit point.

X₃₅₉₃ISOTOMIC CONJUGATE OF 1st DIXIT POINT

X₃₅₉₃ is the isotomic conjugate of 1st dixit point.

X₃₅₉₄ISOGONAL CONJUGATE OF 2nd DIXIT POINT

X₃₅₉₄ is the isogonal conjugate of 2nd dixit point.

X₃₅₉₅ISOTOMIC CONJUGATE OF 2nd DIXIT POINT

X₃₅₉₅ is the isotomic conjugate of 2nd dixit point.

X₃₅₉₆1st ODEHNAL POINT

X₃₅₉₆ = polar conjugate of X₆₀₈

X₃₅₉₇2nd ODEHNAL POINT

Let M_A, M_B, M_C be the centers of the circles A″, B″, C″ used to construct X₃₅₉₆, respectively. Then the lines A-to-M_A, B-to-M_B, C-to-M_C concur in X₃₅₉₇. For more, see the reference and sketch at X₃₅₉₆.

X₃₅₉₈1st LIU POINT

Let O_A be the circle passing through B and C, internally tangent to the incircle, and likewise for O_B and O_C. Let P_A be the point where O_A meets the incircle, and likewise for P_B and P_C. Let Q_A be the point of intersection of the tangents to the incircle at P_B and P_C, and likewise for P_B and P_C. Then X₃₅₉₈ is the perspector of the triangles P_AP_BP_C and Q_AQ_BQ_C; this point is also X₇-of-Q_AQ_BQ_C. This point and X₃₅₉₉ were discovered by Kang-Ying Liu of St. Andrew’s Priory School, Honolulu, Hawaii, during 2010.

X₃₅₉₉2nd LIU POINT

Let P_AP_BP_C be as at X₃₅₉₈. Let T_A be the point of intersection of the lines BP_C and CP_B. X₃₅₉₉ is the perspector of triangles P_AP_BP_C and T_AT_BT_C.

X₃₆₀₀INTERSECTION OF LINES X₁X_7 AND X₈X_57

X₃₆₀₀ was found by Peter Moses in connection with the 1st Liu point.

X₃₆₀₁INTERSECTION OF LINES X₁X_3 AND X₉X_21

X₃₆₀₁, the {X₁, X₃₆₁₁}-harmonic conjugate of X₃₅₇₆, was found by Peter Moses in connection with the 1st Liu point.