X₈₃

CEVAPOINT OF CENTROID AND SYMMEDIAN POINT

X₈₃ is the cevapoint of centroid and symmedian point.

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

X₈₃ (CEVAPOINT OF CENTROID AND SYMMEDIAN POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 32}, {6, 76}

• Related Centers: X₂ (CENTROID), X₆ (SYMMEDIAN POINT), X₃₂ (3rd POWER POINT), X₇₆ (3rd BROCARD POINT)

Trilinears: \(bc/(b^{2} + c^{2}) : ca/(c^{2} + a^{2}) : ab/(a^{2} + b^{2})\)

Barycentrics: \(1/(b^{2} + c^{2}) : 1/(c^{2} + a^{2}) : 1/(a^{2} + b^{2})\)

Search Value (6, 9, 13): \(1.810750169532\)

▸ Alternate Coordinates1 form

• Trilinears: \(\csc(A + \omega ) : \csc(B + \omega ) : \csc(C + \omega )\)

▸ Incident Central Lines14 lines

{2, 32}, {3, 262}, {4, 182}, {5, 98}, {6, 76}, {10, 82}, {17, 624}, {18, 623}, {39, 99}, {213, 239}, {217, 287}, {275, 297}, {597, 671}, {689, 729}

▸ Functional Relations22 relations

• X₈₃ = isogonal conjugate of X₃₉

• X₈₃ = isotomic conjugate of X₁₄₁

• X₈₃ = complement of X₂₈₉₆

• X₈₃ = cevapoint of X₂ and X₆

• X₈₃ = X_i-cross conjugate of X_j for these (i,j): (2, 308), (6, 251), (512, 99)

• X₈₃ = trilinear pole of line X₂₃X₃₈₅ (line is the polar of X₂ wrt the circumcircle, and also the anticomplement of the de Longchamps line, and also the polar of X₅ wrt {circumcircle, nine-point circle}-inverter)

• X₈₃ = crossdifference of every pair of points on line X₆₈₈X₃₀₀₅

• X₈₃ = pole wrt polar circle of trilinear polar of X₄₂₇

• X₈₃ = X₄₈-isoconjugate (polar conjugate) of X₄₂₇

• X₈₃ = perspector of ABC and medial triangle of 1st Brocard triangle

• X₈₃ = crosspoint of X₂ and X₆ wrt both the anticomplementary and tangential triangles

• X₈₃ = trilinear product of vertices of circummedial triangle

• X₈₃ = midpoint of PU(137)

• X₈₃ = bicentric sum of PU(i) for these i: 137, 141

• X₈₃ = homothetic center of 5th anti-Brocard triangle and medial triangle

• X₈₃ = X₈₂₉₀-of-1st-Brocard-triangle

• X₈₃ = perspector of ABC and 1st Brocard triangle of medial triangle

• X₈₃ = perspector of ABC and 1st Brocard triangle of 5th anti-Brocard triangle

• X₈₃ = homothetic center of ABC and cross-triangle of ABC and 5th anti-Brocard triangle

• X₈₃ = Cundy-Parry Phi transform of X₂₆₂

• X₈₃ = Cundy-Parry Psi transform of X₁₈₂

• X₈₃ = barycentric product of circumcircle intercepts of line X₃₁₆X₅₁₂

Notes

Let K denote the symmedian point, X₆. Let Sym_aSym_bSym_c be the cevian triangle of K. Let K_A be K of the triangle ASym_bSym_c; let K_B be K of Sym_aBSym_c and let K_C be K of Sym_aSym_bC. The lines AK_A, BK_B, CK_C concur in X₈₃. (Randy Hutson, 9/23/2011)

Let A'B'C' be the 1st Brocard triangle. Let A″ be the reflection of A' in BC, and define B″ and C″ cyclically. AA″, BB″, CC″ concur in X₈₃. (Randy Hutson, December 26, 2015)

Let B_a, C_a be the intersections of lines CA, AB, resp., and the antiparallel to BC through X₂. 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 X₈₃. (Randy Hutson, December 26, 2015)

Let (O_a) be the circle whose diameter is the orthogonal projections of PU(1) on line BC. Define (O_b) and (O_c) cyclically. X₈₃ is the radical center of circles (O_a), (O_b), (O_c). (Randy Hutson, December 26, 2015)

Let Cm_aCm_bCm_c be the circummedial triangle. Let A″ be the trilinear pole of line Cm_bCm_c, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₈₃. (Randy Hutson, December 26, 2015)