X₉₄₀

POINT ALDEBARAN

X₉₄₀ = crosssum of X₁₁ and X₇₈₄

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

X₉₄₀ (POINT ALDEBARAN) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 3}, {2, 6}

• Related Centers: X₉₄₁ (ISOGONAL CONJUGATE OF X₉₄₀), X₁ (INCENTER), X₃ (CIRCUMCENTER), X₂ (CENTROID), X₆ (SYMMEDIAN POINT)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a^{2} + a(b + c) + 2bc\)

Barycentrics: \(a^{2} + 4 R r : :\)

Search Value (6, 9, 13): \(1.556965023114\)

▸ Alternate Coordinates1 form

• Trilinears: \(as + bc : bs + ca : cs + ab\)

▸ Incident Central Lines12 lines

{1, 3}, {2, 6}, {31, 1001}, {37, 63}, {42, 750}, {58, 405}, {72, 975}, {222, 226}, {312, 894}, {386, 474}, {387, 443}, {518, 612}

▸ Functional Relations8 relations

• X₉₄₀ = isogonal conjugate of X₉₄₁

• X₉₄₀ = isotomic conjugate of X₃₄₂₅₈

• X₉₄₀ = complement of X₅₇₃₉

• X₉₄₀ = anticomplement of X₅₇₄₃

• X₉₄₀ = crosssum of X₁₁ and X₇₈₄

• X₉₄₀ = crossdifference of every pair of points on line X₅₁₂X₆₅₀

• X₉₄₀ = {X₁,X₄₀}-harmonic conjugate of X₃₇₅₄₈

• X₉₄₀ = {X₂,X₆}-harmonic conjugate of X₄₃₈₃

Notes

Let A' be the center of the conic through the contact points of the incircle and the A- excircle with the sidelines of ABC. Define B' and C' cyclically. The triangle A'B'C' is perspective to the 4th extouch triangle at X₉₄₀. See also X₆, X₂₅, X₂₁₈, X₂₂₂, X₁₇₄₃. (Randy Hutson, July 23, 2015)