X₃₂₅₉

CROSSSUM OF X₁₀₀ AND X₁₀₄

A construction see Aris Pavlakis and Peter Moses, Hyacinthos 28741.

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

X₃₂₅₉ (CROSSSUM OF X₁₀₀ AND X₁₀₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Medial Triangle △M_aM_bM_c — vertices M_a, M_b, M_c (Formed by the midpoints of sides BC, CA, AB)

• Central Lines: {2, 901}

• Related Centers: X₂ (CENTROID), X₉₀₁ (ISOGONAL CONJUGATE OF X₉₀₀)

Trilinears: \(1/((a - b)(\cos C + \cos A - 1)) + 1/((c - a)(\cos A + \cos B - 1)) : :\)

Barycentrics: \((2 a - b - c) \cdot (b - c)^{2} (a^{2} b - b^{3} + a^{2} c - 2 a b c + b^{2} c + b c^{2} - c^{3}) : :\)

Search Value (6, 9, 13): \(4.168819137566\)

X₃₂₅₉ lies on the Sherman line (3259,3326); see http://forumgeom.fau.edu/FG2012volume12/FG201220.pdf)

X₃₂₅₉ lies on the nine-point circle, the Yff contact circle.

▸ Incident Central Lines35 lines

{2, 901}, {4, 953}, {11, 513}, {12, 13756}, {25, 10016}, {36, 855}, {56, 17101}, {114, 1281}, {115, 661}, {116, 3835}, {119, 517}, {120, 5087}, {121, 3814}, {133, 5146}, {149, 14513}, {153, 14511}, {226, 24201}, {244, 6615}, {1319, 1846}, {1566, 6544}, {1647, 5516}, {1878, 22835}, {2969, 20620}, {3120, 7336}, {3326, 10017}, {4370, 5513}, {4404, 24026}, {5954, 5993}, {5957, 11792}, {7951, 23153}, {8286, 8819}, {15614, 21252}, {17036, 20096}, {17605, 23152}, {24250, 25760}

▸ Functional Relations21 relations

• X₃₂₅₉ = midpoint of X_i and X_j for these {i,j}: {4, 953}, {56, 17101}, {149, 14513}, {153, 14511}

• X₃₂₅₉ = complement of X₉₀₁

• X₃₂₅₉ = anticomplement of X₂₂₁₀₂

• X₃₂₅₉ = reflection of X_i and X_j for these {i,j}: {901, 22102}, {3937, 14115}, {6073, 119}, {6075, 11}

• X₃₂₅₉ = reflection of X₃₉₃₇ in the line X₁X₃

• X₃₂₅₉ = {X₂,X₉₀₁}-harmonic conjugate of X₂₂₁₀₂

• X₃₂₅₉ = polar circle inverse of X₁₃₀₉

• X₃₂₅₉ = orthoptic circle of the Steiner inellipe inverse of X₂₇₂₆

• X₃₂₅₉ = complement of the isogonal of X₉₀₀

• X₃₂₅₉ = X_i-complementary conjugate of X_j for these (i,j): {1, 900}, {2, 4928}, {6, 3960}, {31, 3310}, {42, 21894}, {44, 514}, {244, 1647}, {513, 519}, {514, 3834}, {519, 513}, {522, 5123}, {649, 16610}, {661, 3936}, {667, 8610}, {678, 6544}, {693, 21241}, {765, 6550}, {876, 25351}, {900, 10}, {902, 650}, {1019, 4395}, {1023, 4422}, {1319, 522}, {1404, 905}, {1635, 2}, {1639, 3452}, {1647, 11}, {1877, 521}, {1960, 37}, {2087, 1086}, {2161, 21198}, {2251, 6586}, {2325, 20317}, {2429, 25097}, {3251, 4370}, {3264, 21260}, {3285, 14838}, {3669, 17067}, {3689, 4521}, {3762, 141}, {3911, 4885}, {3943, 4129}, {4120, 1211}, {4358, 3835}, {4448, 17793}, {4530, 26932}, {4730, 1213}, {4768, 1329}, {4893, 27751}, {4895, 9}, {5440, 20315}, {6544, 16594}, {14407, 16589}, {14408, 6376}, {14427, 6554}, {14429, 21530}, {14437, 13466}, {14584, 3738}, {16704, 4369}, {17780, 24003}, {21805, 661}, {22086, 1214}, {23344, 24036}, {23703, 3035}, {23757, 119}, {23838, 3036}, {24004, 27076}

• X₃₂₅₉ = X_i-Ceva conjugate of X_j for these (i,j): {2, 3310}, {4, 900}, {11, 1647}, {513, 6550}, {1145, 23757}

• X₃₂₅₉ = crosssum of X₁₀₀ and X₁₀₄

• X₃₂₅₉ = crossdifference of every pair of points on the line X₂₄₂₃X₂₄₂₇

• X₃₂₅₉ = crosspoint of X_i and X_j for these (i,j): {513, 517}, {900, 14584}, {1145, 23757}

• X₃₂₅₉ = X_i-isoconjugate of X_j for these (i,j): {104, 9268}, {765, 10428}, {909, 5376}

• X₃₂₅₉ = inverse-in-polar-circle of X₁₃₀₉

• X₃₂₅₉ = perspector of circumconic centered at X₃₃₁₀

• X₃₂₅₉ = orthopole of line X₃X₈

• X₃₂₅₉ = center of circumconic that is locus of trilinear poles of lines passing through X₃₃₁₀; this conic is a rectangular circumhyperbola that is isogonal conjugate of line X₃X₈

• X₃₂₅₉ = barycentric product X_i X_j for these {i,j}: {514, 23757}, {900, 10015}, {908, 1647}, {1086, 1145}, {1769, 3762}, {1846, 26932}, {2087, 3262}, {2397, 6550}, {4120, 23788}, {4530, 22464}, {17205, 21942}

• X₃₂₅₉ = barycentric quotient X_i / X_j for these {i,j}: {517, 5376}, {900, 13136}, {1015, 10428}, {1145, 1016}, {1769, 3257}, {2087, 104}, {2183, 9268}, {2397, 6635}, {2427, 6551}, {2804, 4582}, {3310, 901}, {6550, 2401}, {8661, 2423}, {10015, 4555}, {23757, 190}, {23788, 4615}

Notes

X₃₂₅₉ is the center of the rectangular hyperbola that passes through the points A, B, C, and X₅₆. (Randy Hutson, 9/23/2011)