X₈₆

CEVAPOINT OF INCENTER AND CENTROID

X₈₆ is the cevapoint of incenter and centroid.

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

X₈₆ (CEVAPOINT OF INCENTER AND CENTROID) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 6}, {7, 21}

• Related Centers: X₁ (INCENTER), X₂ (CENTROID), X₆ (SYMMEDIAN POINT), X₇ (GERGONNE POINT), X₂₁ (SCHIFFLER POINT)

Trilinears: \(bc/(b + c) : ca/(c + a) : ab/(a + b)\)

Barycentrics: \(1/(b + c) : 1/(c + a) : 1/(a + b)\)

Search Value (6, 9, 13): \(2.176292659804\)

▸ Linear Combinations1 combination

• X₈₆ = \(2(r^{2} + 2rR + s^{2})\) X₁ + \(3(r^{2} + s^{2})\) X₂ - \(4r^{2}\) X₃ (Peter Moses, April 2, 2013)

▸ Incident Central Lines17 lines

{1, 75}, {2, 6}, {7, 21}, {10, 319}, {29, 34}, {37, 190}, {58, 238}, {60, 272}, {99, 106}, {110, 675}, {142, 284}, {239, 1100}, {269, 1088}, {283, 307}, {310, 350}, {741, 789}, {870, 871}

▸ Functional Relations28 relations

• X₈₆ = isogonal conjugate of X₄₂

• X₈₆ = isotomic conjugate of X₁₀

• X₈₆ = complement of X₁₆₅₄

• X₈₆ = anticomplement of X₁₂₁₃

• X₈₆ = circumcircle-inverse of X₅₉₃₇

• X₈₆ = X₂₇₄-Ceva conjugate of X₃₃₃

• X₈₆ = cevapoint of X_i and X_j for these (i,j): (1, 2), (7, 77), (21, 81)

• X₈₆ = crosssum of X₁ and X₁₀₄₅

• X₈₆ = crossdifference of every pair of points on line X₅₁₂X₇₉₈

• X₈₆ = X_i-cross conjugate of X_j for these (i,j): (1, 81), (2, 274), (7, 286), (21, 333), (58, 27), (513, 190)

• X₈₆ = X_i-beth conjugate of X_j for these (i,j): (86, 1014), (99, 86), (261, 86), (314, 314), (645, 86), (811, 86)

• X₈₆ = X₂-Ceva conjugate of X₆₆₂₆

• X₈₆ = intersection of tangents at X₁ and X₂ to the bianticevian conic of X₁ and X₂; see X₉₉

• X₈₆ = crosspoint of X₁ and X₂ wrt both the excentral and anticomplementary triangles

• X₈₆ = trilinear pole of line X₂₃₉X₅₁₄ (Lemoine axis of excentral triangle)

• X₈₆ = pole wrt polar circle of trilinear polar of X₁₈₂₆

• X₈₆ = X₄₈-isoconjugate (polar conjugate)-of-X₁₈₂₆

• X₈₆ = perspector of Gemini triangle 1 and cross-triangle of ABC and Gemini triangle 1

• X₈₆ = perspector of ABC and cross-triangle of ABC and Gemini triangle 23

• X₈₆ = perspector of ABC and cross-triangle of ABC and Gemini triangle 24

• X₈₆ = perspector of ABC and cross-triangle of Gemini triangles 23 and 24

• X₈₆ = perspector of ABC and Gemini triangle 25

• X₈₆ = barycentric product of vertices of Gemini triangle 23

• X₈₆ = barycentric product of vertices of Gemini triangle 24

• X₈₆ = barycentric product of vertices of Gemini triangle 25

• X₈₆ = {X₂,X₆}-harmonic conjugate of X₁₇₂₇₇

• X₈₆ = {X₂,X₆₉}-harmonic conjugate of X₅₂₂₄

• X₈₆ = {X₂,X₁₄₁}-harmonic conjugate of X₁₇₃₀₇

Notes

Let A'B'C' be the anticomplement of the Feuerbach triangle. Let L_a be the tangent to the circumcircle at A', and define L_b and L_c cyclically. Let A″ be the point where L_a is tangent to the Steiner circumellipse, and define B″ and C″ cyclically. Let A* = BB″∩CC″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X₈₆. (Randy Hutson, December 10, 2016)

Let A₁B₁C₁ be the 1st Conway triangle. Let A' be the trilinear pole of line B₁C₁, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₈₆. (Randy Hutson, December 10, 2016)

Let A₅B₅C₅ be the 5th Conway triangle. Let A' be the trilinear pole of line B₅C₅, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₈₆. (Randy Hutson, December 10, 2016)

Let A₃B₃C₃ and A₄B₄C₄ be Gemini triangles 3 and 4, resp. Let L_A be the tangent at A to conic {A,B₃,C₃,B₄,C₄}}, and define L_B, L_C cyclically. Let A' = L_B∩L_C, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₈₆. (Randy Hutson, January 15, 2019)

Let A₂₁B₂₁C₂₁ be Gemini triangle 21. Let A' be the perspector of conic {A, B, C,B₂₁,C₂₁}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₈₆. (Randy Hutson, January 15, 2019)