X₂₄₃

X₄-LINE CONJUGATE OF X₇₃

X₂₄₃ = polar-circle-inverse of X₂₂₆

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

X₂₄₃ (X₄-LINE CONJUGATE OF X₇₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 4}, {55, 92}

• Related Centers: X₄ (ORTHOCENTER), X₇₃ (CROSSPOINT OF INCENTER AND CIRCUMCENTER), X₁ (INCENTER), X₅₅ (INSIMILICENTER), X₉₂ (CEVAPOINT OF INCENTER AND CLAWSON POINT)

Trilinears: \((\sec A)(\cos^{2} A - \cos B \cos C) : :\)

Barycentrics: \((a - b - c) (a^{4} - a^{2} (b^{2} - b c + c^{2}) - b c (b - c)^{2})/(a^{2} - b^{2} - c^{2}) : :\)

Search Value (6, 9, 13): \(1.882963431523\)

X₂₄₃ lies on the cubics K040, K683, K1185.

▸ Incident Central Lines103 lines

{1, 4}, {2, 1857}, {3, 158}, {11, 17923}, {20, 1118}, {21, 1896}, {27, 1859}, {29, 2646}, {36, 1784}, {46, 1148}, {55, 92}, {56, 1895}, {57, 43160}, {65, 412}, {90, 7049}, {105, 1309}, {107, 14192}, {108, 2723}, {132, 8229}, {162, 2361}, {196, 3474}, {212, 1957}, {240, 7004}, {242, 1283}, {281, 5218}, {296, 8764}, {318, 958}, {411, 821}, {415, 2652}, {425, 662}, {468, 42069}, {518, 1897}, {522, 652}, {648, 2651}, {653, 1155}, {851, 41500}, {917, 2222}, {920, 1075}, {929, 20624}, {1040, 1096}, {1324, 14017}, {1430, 1936}, {1447, 47212}, {1465, 36127}, {1503, 38357}, {1715, 7098}, {1788, 3176}, {1837, 5125}, {1854, 5786}, {1858, 2907}, {1861, 26013}, {1864, 37279}, {1871, 12711}, {1875, 37420}, {1882, 7513}, {1944, 15146}, {1948, 7360}, {2073, 14194}, {2202, 39032}, {2376, 2731}, {2586, 34592}, {2587, 34593}, {2659, 37142}, {3100, 14956}, {3147, 7040}, {3149, 47372}, {3326, 5057}, {3601, 39585}, {3612, 7531}, {3685, 36797}, {3812, 37278}, {4219, 40149}, {4459, 15150}, {5081, 44669}, {5174, 10950}, {5205, 6335}, {5219, 39531}, {5327, 41083}, {5794, 17555}, {6331, 14195}, {6336, 14190}, {6985, 20764}, {7017, 7081}, {7288, 40836}, {8748, 40937}, {8758, 43764}, {9371, 14571}, {10589, 17917}, {11502, 35994}, {13149, 14189}, {13411, 39574}, {14006, 41234}, {14202, 17983}, {15252, 51368}, {17605, 42387}, {22768, 37253}, {23207, 35981}, {23353, 26884}, {23711, 37769}, {24032, 36002}, {24929, 39529}, {36599, 38249}, {37737, 44225}

▸ Functional Relations27 relations

• X₂₄₃ = reflection of X₅₁₃₆₈ in X₁₅₂₅₂

• X₂₄₃ = isogonal conjugate of X₂₉₆

• X₂₄₃ = incircle-inverse of X₄₀₉₆₀

• X₂₄₃ = polar-circle-inverse of X₂₂₆

• X₂₄₃ = polar conjugate of X₁₉₅₂

• X₂₄₃ = polar conjugate of the isotomic conjugate of X₁₉₄₄

• X₂₄₃ = polar conjugate of the isogonal conjugate of X₁₉₅₁

• X₂₄₃ = orthic-isogonal conjugate of X₄₁₄₉₉

• X₂₄₃ = X₃₁-complementary conjugate of X₃₉₀₃₃

• X₂₄₃ = X_i-Ceva conjugate of X_j for these (i,j): {2, 39033}, {4, 41499}, {1937, 1940}, {8764, 1}, {15146, 1936}, {41207, 650}, {43764, 4}

• X₂₄₃ = X_i-cross conjugate of X_j for these (i,j): {851, 1936}, {1951, 1944}

• X₂₄₃ = cevapoint of X₄₁₅ and X₂₆₅₉

• X₂₄₃ = crosspoint of X_i and X_j for these (i,j): {1, 2656}, {1937, 7105}

• X₂₄₃ = crosssum of X_i and X_j for these (i,j): {1, 2655}, {3, 17975}, {1935, 1936}

• X₂₄₃ = crossdifference of every pair of points on line {73, 652}

• X₂₄₃ = X_i-lineconjugate of X_j for these (i,j): {1, 73}, {522, 652}

• X₂₄₃ = X_i-Hirst inverse of X_j for these (i,j): (1, 4), (46, 1148)

• X₂₄₃ = perspector of conic {A, B, C,X₂₉,X₆₅₃,PU(15)}}

• X₂₄₃ = perspector of conic {A, B, C,X₂₉,X₆₅₃,PU(15)}

• X₂₄₃ = crossdifference of PU(16)

• X₂₄₃ = pole wrt polar circle of the line X₂₂₆X₅₂₂

• X₂₄₃ = X₄₈-isoconjugate (polar conjugate) of X₁₉₅₂

• X₂₄₃ = X_i-isoconjugate of X_j for these (i,j): {1, 296}, {2, 1949}, {3, 1937}, {6, 40843}, {48, 1952}, {63, 1945}, {73, 37142}, {647, 41206}, {822, 41207}, {1214, 2249}, {1409, 35145}, {1935, 1942}

• X₂₄₃ = X_i-Dao conjugate of X_j for these (i,j): {2, 39033}, {3, 296}, {3, 39037}, {9, 40843}, {63, 39032}, {69, 39035}, {75, 39036}, {307, 35075}, {1249, 1952}, {1937, 36103}, {1943, 39034}, {1945, 3162}, {1949, 32664}, {39052, 41206}

• X₂₄₃ = barycentric product X_i*X_j for these {i,j}: {1, 1948}, {4, 1944}, {29, 8680}, {75, 2202}, {92, 1936}, {158, 6518}, {226, 15146}, {264, 1951}, {278, 7360}, {281, 5088}, {312, 1430}, {450, 7108}, {522, 1981}, {851, 31623}, {1172, 44150}, {2656, 39036}, {4391, 23353}, {7017, 26884}, {15418, 18344}, {17947, 41499}, {40843, 41500}, {42669, 44130}

• X₂₄₃ = barycentric quotient X_i/X_j for these {i,j}: {1, 40843}, {4, 1952}, {6, 296}, {19, 1937}, {25, 1945}, {29, 35145}, {31, 1949}, {107, 41207}, {162, 41206}, {450, 1943}, {851, 1214}, {1172, 37142}, {1430, 57}, {1936, 63}, {1944, 69}, {1948, 75}, {1951, 3}, {1981, 664}, {2202, 1}, {2299, 2249}, {5088, 348}, {6518, 326}, {7106, 1942}, {7360, 345}, {8680, 307}, {15146, 333}, {23353, 651}, {26884, 222}, {41368, 1940}, {41497, 1947}, {41499, 17950}, {41500, 1948}, {42669, 73}, {44096, 1950}, {44112, 1409}, {44150, 1231}, {51647, 1439}

• X₂₄₃ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {1, 2655, 8763}, {1, 51282, 4}, {3, 158, 1940}, {278, 44695, 497}, {1309, 16082, 14198}, {2635, 8763, 2655}, {2646, 42385, 29}, {3176, 37417, 1788}

Notes

X₂₄₃ is the perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C, and the pair P(15) and U(15) of bicentric points (see the notes just before X₁₉₀₈. (Randy Hutson, 9/23/2011)

X₂₄₃ is the point of intersection of the following lines:

• trilinear polars of P(15) and U(15)

• X₁X₄

• trilinear polar of cevapoint{X₁,X₄}

• UV, where U = X₁-Ceva-conjugate-of-(4) = X₁₁₄₈, and V = X₄-Ceva-conjugate-of-X₁ = X₄₆ (Randy Hutson, December 26, 2015)