X₂₂₃

X₂-CEVA CONJUGATE OF X₅₇

X₂₂₃ = complement of X₁₈₉

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

X₂₂₃ (X₂-CEVA CONJUGATE OF X₅₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 57}, {2, 77}

• Related Centers: X₂ (CENTROID), X₅₇ (ISOGONAL CONJUGATE OF X₉), X₆ (SYMMEDIAN POINT), X₇₇ (ISOGONAL CONJUGATE OF X₃₃)

Trilinears: \((\tan A/2)(\cos B + \cos C - \cos A - 1)\)

Search Value (6, 9, 13): \(-0.800760299150\)

▸ Alternate Coordinates2 forms

• Trilinears: \(\sec B + \sec C - \sec A + 1 : :\)

• Trilinears: \((a^{3} + a^{2} b - a b^{2} - b^{3} + a^{2} c - 2 a b c + b^{2} c - a c^{2} + b c^{2} - c^{3})/(b + c - a) : :\)

X₂₂₃ lies on the Thomson cubic.

▸ Incident Central Lines18 lines

{2, 77}, {3, 1035}, {6, 57}, {9, 1073}, {40, 221}, {55, 1456}, {56, 937}, {63, 651}, {108, 204}, {109, 165}, {312, 664}, {329, 347}, {380, 608}, {580, 603}, {936, 1038}, {1249, 3352}, {3341, 3349}, {3351, 3356}

▸ Functional Relations23 relations

• X₂₂₃ = isogonal conjugate of X₂₈₂

• X₂₂₃ = isotomic conjugate of X₃₄₄₀₄

• X₂₂₃ = complement of X₁₈₉

• X₂₂₃ = anticomplement of X₂₀₂₀₅

• X₂₂₃ = complementary conjugate of X₂₁₂₃₉

• X₂₂₃ = X_i-Ceva conjugate of X_j for these (i,j): (2, 57), (77, 1), (342, 208), (347, 40)

• X₂₂₃ = cevapoint of X₁₉₈ and X₂₂₁

• X₂₂₃ = X_i-cross conjugate of X_j for these (i,j): (198, 40), (227, 347)

• X₂₂₃ = crosspoint of X₂ and X₃₂₉

• X₂₂₃ = crosssum of X₆ and X₁₄₃₆

• X₂₂₃ = perspector of ABC and antipedal triangle of X₃₃₄₅

• X₂₂₃ = perspector of pedal and anticevian triangles of X₃₁₈₂

• X₂₂₃ = perspector of ABC and medial triangle of pedal triangle of X₁₄₉₀

• X₂₂₃ = perspector of circumconic centered at X₅₇

• X₂₂₃ = center of circumconic that is locus of trilinear poles of lines passing through X₅₇

• X₂₂₃ = X₉₂-isoconjugate of X₂₁₈₈

• X₂₂₃ = {X₁,X₁₇₄₅}-harmonic conjugate of X₁₄₉₀

• X₂₂₃ = pole wrt polar circle of trilinear polar of X₇₀₂₀

• X₂₂₃ = X₄₈-isoconjugate (polar conjugate) of X₇₀₂₀

• X₂₂₃ = trilinear product X₄₀*X₅₇

• X₂₂₃ also lies on line 55,1456.

• X₂₂₃ = X_i-aleph conjugate of X_j for these (i,j): (63, 1079), (77, 223), (81, 580), (174, 46), (651, 109)

• X₂₂₃ = X_i-beth conjugate of X_j for these (i,j): (2, 278), (100, 200), (162, 204), (329, 329), (651, 223), (662, 63)

Notes

Let A' be the homothetic center of ABC and the orthic triangle of the A-extouch triangle and define B' and C' cyclically. Triangle A'B'C' is perspective to the 3rd extouch triangle at X₂₂₃. (Randy Hutson, September 14, 2016)

Let V_a, V_b, V_c be the antipodes of V=X₄₀ in the circles (VBC), (VCA), (VAB), respectively. The triangle V_aV_bV_c is here named the Bevan antipodal triangle (Hyacinthos #29638; however, see next note). The point X₂₂₃ is the unique finite fixed point of the affine transformation that maps the reference triangle ABC onto V_aV_bV_c. (Angel Montesdeoca, October 15, 2019)

The construction of the triangle V_aV_bV_c can be generalized for an arbitrary point P (instead of V=X₄₀), and is, in fact, the antipedal triangle of P (Randy Hutson, message in Anopolis Group). Thus, “Bevan antipodal triangle” = antipedal triangle of X₄₀. (Angel Montesdeoca, October 16, 2019)