X₈₁

CEVAPOINT OF INCENTER AND SYMMEDIAN POINT

X₈₁ is the cevapoint of incenter and symmedian point.

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

X₈₁ (CEVAPOINT OF INCENTER AND SYMMEDIAN POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 6}, {1, 21}

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

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

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

Search Value (6, 9, 13): \(1.333662438745\)

▸ Alternate Coordinates1 form

• Barycentrics: eccentricity of A-Soddy ellipse : :

▸ Linear Combinations1 combination

• X₈₁ = \((r^{2} + 2rR + s^{2})\) X₁ - \(3rR\) X₂ - \(2r^{2}\) X₃ (Peter Moses, April 2, 2013)

▸ Incident Central Lines32 lines

{1, 21}, {2, 6}, {7, 27}, {8, 1010}, {19, 969}, {28, 60}, {29, 189}, {32, 980}, {42, 100}, {43, 750}, {55, 1002}, {56, 959}, {57, 77}, {65, 961}, {88, 662}, {99, 739}, {105, 110}, {145, 1043}, {226, 651}, {239, 274}, {314, 321}, {377, 387}, {386, 404}, {411, 581}, {593, 757}, {715, 932}, {859, 957}, {941, 967}, {982, 985}, {1019, 1022}, {1051, 1054}, {1098, 1104}

▸ Functional Relations28 relations

• X₈₁ = isogonal conjugate of X₃₇

• X₈₁ = isotomic conjugate of X₃₂₁

• X₈₁ = anticomplement of X₁₂₁₁

• X₈₁ = circumcircle-inverse of X₅₈₆₇

• X₈₁ = X_i-Ceva conjugate of X_j for these (i,j): (7, 229), (86, 21), (286, 28)

• X₈₁ = cevapoint of X_i and X_j for these (i,j): (1, 6), (57, 222), (58, 284)

• X₈₁ = X_i-cross conjugate of X_j for these (i,j): (1, 86), (3, 272), (6, 58), (57, 27), (284, 21)

• X₈₁ = crosspoint of X₂₇₄ and X₂₈₆

• X₈₁ = crosssum of X_i and X_j for these (i,j): (1, 846), (6, 1030), (42, 1334), (213, 228)

• X₈₁ = crossdifference of every pair of points on line X₅₁₂X₆₆₁

• X₈₁ = X_i-beth conjugate of X_j for these (i,j): (333, 333), (643, 81), (645, 81), (648, 81), (662, 81), (931, 81)

• X₈₁ = trilinear product of PU(31)

• X₈₁ = intersection of tangents at X₁ and X₆ to the Stammler hyperbola

• X₈₁ = crosspoint of X₁ and X₆ wrt both the excentral and tangential triangles

• X₈₁ = trilinear pole of line X₃₆X₂₃₈ (the polar of X₁ wrt the circumcircle)

• X₈₁ = {X₁,X₃₁}-harmonic conjugate of X₁₆₂₁

• X₈₁ = X₆-isoconjugate of X₁₀

• X₈₁ = X₉₂-isoconjugate of X₂₂₈

• X₈₁ = perspector of ABC and cross-triangle of Gemini triangles 1 and 2

• X₈₁ = barycentric product of vertices of Gemini triangle 1

• X₈₁ = barycentric product of vertices of Gemini triangle 2

• X₈₁ = barycentric product of vertices of Gemini triangle 3

• X₈₁ = barycentric product of vertices of Gemini triangle 4

• X₈₁ = perspector of Gemini triangles 2 and 7

• X₈₁ = perspector of ABC and cross-triangle of ABC and Gemini triangle 1

• X₈₁ = perspector of ABC and cross-triangle of ABC and Gemini triangle 2

• X₈₁ = perspector of Gemini triangle 24 and cross-triangle of ABC and Gemini triangle 24

• X₈₁ = perspector of Gemini triangle 28 and cross-triangle of ABC and Gemini triangle 28

Notes

Let I_aI_bI_c be the cevian triangle of X₁. Let A″ be the symmedian point of triangle AI_bI_c, and define B″ and C″ cyclically. Then the lines AA″, BB″, CC″ concur in X₈₁. (Eric Danneels, Hyacinthos 7892, 9/13/03)

Let I_aI_bI_c be the incentral triangle. Let L_A be the reflection of I_bI_c in the internal angle bisector of vertex angle A, and define L_B and L_C cyclically. Let A″ = L_B∩L_C, B″ = L_C∩L_A, C″ = L_A∩L_B. The lines AA″, BB″, CC″ concur in X₈₁. (Randy Hutson, 9/23/2011)

Let H* be the Stammler hyperbola. Let Tan_aTan_bTan_c be the tangential triangle and J_aJ_bJ_c be the excentral triangle. Let A* be the intersection of the tangents to H* at Tan_a and J_a, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X₈₁. (Randy Hutson, February 10, 2016)

Let A'B'C' be the 2nd circumperp triangle. Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₈₁. (Randy Hutson, February 10, 2016)

Let A'B'C' be the anticomplement of the Feuerbach triangle. Let A″ be BB'∩CC', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₈₁. (Randy Hutson, February 10, 2016)

Let B_a, C_a be the intersections of lines CA, AB, resp., and the antiparallel to BC through X₁. Define C_b, A_b, A_c, B_c cyclically. Triangles AB_aC_a, A_bBC_b, A_cB_cC are similar to each other and inversely similar to ABC. Let S_a be the similitude center of triangles A_bBC_b and A_cB_cC. Define S_b and S_c cyclically. The lines AS_a, BS_b, CS_c concur in X₈₁. (Randy Hutson, February 10, 2016)

Let A₁₀B₁₀C₁₀ be Gemini triangle 10. Let A' be the perspector of conic {A, B, C,B₁₀,C₁₀}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₈₁. (Randy Hutson, January 15, 2019)

Let A₁₁B₁₁C₁₁ be Gemini triangle 11. Let A' be the perspector of conic {A, B, C,B₁₁,C₁₁}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₈₁. (Randy Hutson, January 15, 2019)