X₁₂₇₈

CONGRUENT MIDWAY-PARALLELIANS POINT

X₁₂₇₈ is the {X₇₅,X₁₉₂}-harmonic conjugate of X₂. For a list of harmonic conjugates, click Tables at the top of this page.

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

X₁₂₇₈ (CONGRUENT MIDWAY-PARALLELIANS POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {192, 75}

• Related Centers: X₁₉₂ (-CEVA CONJUGATE OF X), X₇₅ (ISOTOMIC CONJUGATE OF INCENTER)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc(ab +ac - 3bc)\)

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(6.937732115121\)

▸ Incident Central Lines4 lines

{2, 37}, {8, 726}, {145, 740}, {193, 742}

▸ Functional Relations8 relations

• X₁₂₇₈ = reflection of X₁₉₂ in X₇₅

• X₁₂₇₈ = isogonal conjugate of X₃₆₆₁₄

• X₁₂₇₈ = isotomic conjugate of X₃₈₂₄₇

• X₁₂₇₈ = crosssum of X_i and X_j for these {i,j}: {512, 23571}, {649, 23470}, {667, 23560}

• X₁₂₇₈ = crossdifference of every pair of points on line X₆₆₇X₂₃₄₇₂

• X₁₂₇₈ = anticomplement of X₁₉₂

• X₁₂₇₈ = complement of X₄₇₈₈

• X₁₂₇₈ = polar conjugate of isogonal conjugate of X₂₂₁₄₉

Notes

Antreas Hatzipolakis described this point in Hyacinthos #3190 (7/13/2001). See also

Paul Yiu, Introduction to the Geometry of the Triangle, 2002, Section 3.3.1, exercise 2.