X₃₂₁₈

INTERSECTION OF LINES X₂X_7 AND X₈X_46

X₃₂₁₈ = anticomplement of X₉₀₈

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

X₃₂₁₈ (INTERSECTION OF LINES X₂X_7 AND X₈X_46) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 7}, {8, 46}

• Related Centers: X₂₁₆₁ (-ISOCONJUGATE OF X₃₆), X₂ (CENTROID), X₇ (GERGONNE POINT), X₈ (NAGEL POINT), X₄₆ (-CEVA CONJUGATE OF X₁)

Trilinears: \(a^{2} - b^{2} - c^{2} + bc : :\)

Search Value (6, 9, 13): \(-12.018005004412\)

▸ Alternate Coordinates1 form

• Trilinears: \((1 - 2 \cos A) \csc A : :\) (M. Iliev, 4/12/07)

▸ Incident Central Lines55 lines

{1, 89}, {2, 7}, {8, 46}, {11, 1776}, {21, 942}, {22, 1473}, {31, 982}, {36, 214}, {37, 2666}, {38, 171}, {40, 145}, {44, 88}, {65, 2975}, {72, 404}, {75, 1150}, {80, 535}, {81, 593}, {97, 1214}, {100, 518}, {104, 517}, {110, 2651}, {149, 516}, {153, 1512}, {189, 2994}, {191, 1125}, {222, 1993}, {229, 1098}, {238, 244}, {239, 514}, {240, 1430}, {241, 1252}, {278, 1748}, {291, 2239}, {320, 2245}, {323, 1443}, {324, 1947}, {333, 2160}, {335, 675}, {354, 1621}, {388, 1454}, {394, 1407}, {411, 1071}, {484, 519}, {614, 1707}, {651, 1465}, {750, 984}, {799, 1921}, {899, 1054}, {903, 2161}, {919, 1814}, {962, 1158}, {986, 1468}, {990, 2000}, {1012, 2095}, {1046, 1193}

▸ Functional Relations15 relations

• X₃₂₁₈ = reflection of X_i in X_j for these (i,j): (100, 1155), (153, 1512), (3257, 44)

• X₃₂₁₈ = isogonal conjugate of X₂₁₆₁

• X₃₂₁₈ = isotomic conjugate of X₁₈₃₅₉

• X₃₂₁₈ = complement of X₁₇₄₈₄

• X₃₂₁₈ = anticomplement of X₉₀₈

• X₃₂₁₈ = X₉₀₃-Ceva conjugate of X₁

• X₃₂₁₈ = cevapoint of X₃₆ and X₂₃₂₃

• X₃₂₁₈ = X_i-cross conjugate of X_j for these (i,j): (36, 1443), (2245, 36)

• X₃₂₁₈ = crosspoint of X₈₁ and X₈₈

• X₃₂₁₈ = crosssum of X_i and X_j for these (i,j): (37, 44), (649, 2087), (1635, 2170)

• X₃₂₁₈ = crossdifference of every pair of points on the line X₄₂X₆₆₃

• X₃₂₁₈ = inverse-in-circumconic-centered-at-X₉ of X₂

• X₃₂₁₈ = {X₉,X₅₇}-harmonic conjugate of X₃₃₀₆

• X₃₂₁₈ = X₁₄₉₅-of-excentral-triangle

• X₃₂₁₈ = endo-homothetic center of X₃- and X₄-Ehrmann triangles

Notes

This point occurs in a Sadi Abu-Saymeh, Mowaffaq Hajja, and Hellmuth Stachel, “Another cubic associated with the triangle,” Journal for Geometry and Graphics 11 (2007) 15-26.

Let L_a, L_b, L_c be the lines through A, B, C, resp. parallel to the antiorthic axis. Let M_a, M_b, M_c be the reflections of BC, CA, AB in L_a, L_b, L_c, resp. Let A' = M_b∩M_c, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A″B″C″ be the reflection of A'B'C' in the antiorthic axis. The triangle A″B″C″ is homothetic to ABC, with center of homothety X₉₀₈ and centroid X₃₂₁₈; see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)