X₄₈

CROSSPOINT OF X₁ AND X₆₃

X₄₈ is the crosspoint of X₁ and X₆₃.

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

X₄₈ (CROSSPOINT OF X₁ AND X₆₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 19}, {6, 41}

• Related Centers: X₁ (INCENTER), X₆₃ (ISOGONAL CONJUGATE OF X₁₉), X₁₉ (CLAWSON POINT), X₆ (SYMMEDIAN POINT), X₄₁ (-CEVA CONJUGATE OF X₃₁)

Trilinears: \(\tan B + \tan C : :\)

Search Value (6, 9, 13): \(16.295130013943\)

▸ Alternate Coordinates4 forms

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

• Trilinears: \(S_{B}\)\(S_{C}\) - \(S^{2}\) : \(S_{C}\)\(S_{A}\) - \(S^{2}\) : \(S_{A}\)\(S_{B}\) - \(S^{2}\)

• Trilinears: \(1 - \cot B \cot C : :\)

• Trilinears: \(\sin 2A : :\)

▸ Linear Combinations1 combination

• X₄₈ = \((r^{2} + 4rR + 4R^{2} + s^{2})\) X₁ - \(6R(2R + r)\) X₂ - \(2(r^{2} + 2rR - s^{2})\) X₃ (Peter Moses, April 2, 2013)

▸ Incident Central Lines22 lines

{1, 19}, {3, 71}, {6, 41}, {9, 101}, {31, 560}, {36, 579}, {37, 205}, {42, 197}, {55, 154}, {63, 326}, {75, 336}, {163, 1094}, {184, 212}, {220, 963}, {255, 563}, {281, 944}, {282, 947}, {354, 584}, {577, 603}, {692, 911}, {949, 1037}, {958, 965}

▸ Functional Relations23 relations

• X₄₈ = isogonal conjugate of X₉₂

• X₄₈ = isotomic conjugate of X₁₉₆₉

• X₄₈ = complement of X₂₁₂₇₀

• X₄₈ = anticomplement of X₂₀₃₀₅

• X₄₈ = X_i-Ceva conjugate of X_j for these (i,j): (1, 31), (2, 36033), (3, 212), (63, 255), (92, 47), (284, 6)

• X₄₈ = X₂₂₈-cross conjugate of X₃

• X₄₈ = crosspoint of X_i and X_j for these (i,j): (1, 63), (3, 222), (91, 92), (219, 268)

• X₄₈ = crosssum of X_i and X_j for these (i,j): (1, 19), (4, 281), (47, 48), (196, 278), (523, 1146), (661, 1109)

• X₄₈ = crossdifference of every pair of points on line X₂₄₀X₅₂₂

• X₄₈ = X₁-line conjugate of X₂₄₀

• X₄₈ = X_i-beth conjugate of X_j for these (i,j): (101, 48), (219, 219), (284, 604), (906, 48)

• X₄₈ = barycentric product of PU(16)

• X₄₈ = vertex conjugate of PU(18)

• X₄₈ = bicentric sum of PU(22)

• X₄₈ = PU(22)-harmonic conjugate of X₆₅₆

• X₄₈ = trilinear pole of line X₈₁₀X₈₂₂

• X₄₈ = X₂-isoconjugate of X₄

• X₄₈ = X₇₅-isoconjugate of X₁₉

• X₄₈ = X₉₁-isoconjugate of X₁₇₄₈

• X₄₈ = perspector of ABC and extraversion triangle of X₄₈ (which is also the anticevian triangle of X₄₈)

• X₄₈ = crosspoint of X₂₀₆₆ and X₅₄₁₄

• X₄₈ = {X₁,X₁₉}-harmonic conjugate of X₁₉₅₃

• X₄₈ = perspector of circumconic centered at X₃₆₀₃₃

Notes

Let A'B'C' be the hexyl triangle. Let A″ be the barycentric product of the circumcircle intercepts of line B'C'. Define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X₄₈. (Randy Hutson, July 31 2018)

X₄₈ is the {X₄₁,X₆₀₄}-harmonic conjugate of X₆.