X₂₃₇

X₃-LINE CONJUGATE OF X₂

X₂₃₇ is the point of intersection of the Euler line and the Lemoine axis (defined as the radical axis of the circumcircle and the Brocard circle).

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

X₂₃₇ (X₃-LINE CONJUGATE OF X₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {39, 51}

• Related Centers: X₃ (CIRCUMCENTER), X₂ (CENTROID), X₃₉ (BROCARD MIDPOINT), X₅₁ (CENTROID OF ORTHIC TRIANGLE)

Trilinears: \(a^{2}\cos(A + \omega ) : b^{2}\cos(B + \omega ) : c^{2}\cos(C + \omega )\)

Barycentrics: \(a^{3}\cos(A + \omega ) : b^{3}\cos(B + \omega ) : c^{3}\cos(C + \omega )\)

Search Value (6, 9, 13): \(-1.216425668840\)

▸ Alternate Coordinates2 forms

• Trilinears: \(a^{3}(b^{4} + c^{4} - a^{2}b^{2} - a^{2}c^{2})\) (Darij Grinberg, 3/29/03)

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

▸ Linear Combinations1 combination

• X₂₃₇ = X₂₃₇ = \(3\) X₂ + X₄₆₅₁₈, \(2\) X₃₀₀₁ - \(3\) X₂₂₀₈₇, \(3\) X₉₁₅₅ - \(2\) X₃₆₂₁₂, X₁₄₉₅₇ - \(6\) X₄₄₂₁₅, X₂₁₅₃₁ - \(3\) X₄₄₂₁₅, \(2\) X₂₁₅₃₁ + X₄₆₅₁₈, \(3\) X₂₂₀₈₇ - \(4\) X₃₄₉₉₀, X₃₄₁₇₅ - \(3\) X₃₈₂₂₇, \(6\) X₄₄₂₁₅ + X₄₆₅₁₈

X₂₃₇ lies on the cubics K380, K532, K782, K783, K789, K1119.

▸ Incident Central Lines201 lines

{2, 3}, {6, 160}, {15, 14186}, {16, 14188}, {31, 904}, {32, 184}, {35, 40790}, {39, 51}, {42, 18758}, {50, 1576}, {55, 20471}, {95, 32085}, {98, 6037}, {99, 3978}, {110, 2080}, {111, 46316}, {141, 8266}, {154, 682}, {157, 8553}, {159, 1609}, {187, 351}, {193, 20794}, {206, 571}, {216, 1843}, {232, 9475}, {248, 1971}, {323, 9301}, {325, 20022}, {373, 21163}, {385, 3511}, {476, 43654}, {511, 9155}, {524, 1634}, {566, 9971}, {570, 9969}, {574, 34417}, {577, 1974}, {672, 20777}, {694, 1691}, {800, 6467}, {804, 46302}, {805, 39092}, {1030, 22369}, {1084, 33875}, {1194, 13357}, {1284, 43920}, {1384, 9463}, {1503, 20021}, {1624, 15448}, {1660, 40320}, {1661, 40321}, {1755, 5360}, {1899, 8721}, {1915, 34870}, {1975, 20023}, {2021, 3124}, {2076, 34214}, {2183, 23198}, {2187, 18756}, {2211, 36425}, {2269, 22389}, {2351, 2353}, {2352, 20284}, {2393, 3003}, {2482, 38998}, {2936, 9890}, {2979, 9821}, {2980, 45838}, {3001, 9019}, {3049, 43112}, {3060, 3095}, {3164, 3186}, {3284, 44102}, {3289, 9418}, {3313, 20819}, {3398, 5012}, {3564, 25046}, {3589, 35222}, {3785, 14826}, {3917, 5188}, {3926, 9917}, {3964, 37491}, {5007, 13366}, {5008, 44109}, {5013, 17810}, {5023, 21001}, {5024, 13192}, {5041, 34565}, {5063, 19136}, {5065, 33578}, {5092, 34236}, {5104, 33876}, {5158, 8541}, {5162, 37841}, {5171, 9306}, {5206, 44082}, {5210, 41424}, {5467, 32217}, {5640, 11171}, {5651, 8722}, {5938, 15340}, {5943, 13334}, {5976, 8840}, {6000, 44437}, {6324, 13531}, {6403, 30258}, {6530, 19189}, {6752, 44088}, {7113, 22096}, {7669, 11063}, {7716, 36751}, {7772, 15004}, {7782, 41259}, {7795, 43653}, {8265, 41331}, {8299, 20878}, {8570, 10329}, {8573, 19459}, {8617, 15655}, {8705, 46127}, {8724, 15360}, {8880, 12049}, {8881, 12048}, {9142, 12367}, {9149, 22329}, {9157, 32526}, {9468, 32748}, {9512, 44375}, {9605, 9777}, {9752, 22655}, {10132, 39648}, {10133, 39679}, {10313, 44089}, {10316, 23606}, {10317, 23357}, {11002, 32447}, {11003, 11842}, {11257, 39906}, {11402, 30435}, {11649, 18114}, {12143, 26166}, {12215, 24729}, {14251, 23611}, {14547, 23199}, {14601, 32654}, {14908, 32640}, {14961, 44084}, {14981, 41586}, {15066, 34095}, {15080, 26316}, {15107, 33873}, {15166, 44125}, {15167, 44126}, {15624, 17454}, {15905, 19118}, {16872, 23868}, {16985, 39652}, {17984, 18024}, {18265, 18266}, {18755, 23212}, {18860, 23217}, {19121, 37893}, {19599, 25332}, {20080, 22152}, {20968, 44078}, {21352, 37575}, {21639, 40135}, {22052, 44091}, {22143, 37784}, {22401, 44079}, {23098, 34157}, {23158, 41588}, {23181, 32269}, {23197, 40956}, {25054, 44371}, {32224, 40879}, {32428, 44145}, {32564, 39649}, {32571, 39658}, {32713, 44096}, {33581, 40319}, {33582, 44200}, {33871, 46327}, {33972, 34013}, {34175, 38227}, {34566, 34571}, {35007, 44110}, {35383, 38873}, {36822, 46777}, {37479, 43650}, {37512, 44106}, {38354, 41167}, {38383, 46807}, {39557, 40850}, {41005, 41584}, {41196, 44778}, {41197, 44779}, {42329, 43976}, {44192, 45429}, {44193, 45428}, {46286, 46306}

▸ Functional Relations36 relations

• X₂₃₇ = midpoint of X_i and X_j for these {i,j}: {23, 7468}, {1634, 5201}, {14957, 46518}

• X₂₃₇ = reflection of X_i in X_j for these {i,j}: {2, 44215}, {3, 44221}, {4, 44227}, {3001, 34990}, {14957, 21531}, {20975, 3003}, {36189, 468}, {38368, 232}, {46522, 21177}

• X₂₃₇ = isogonal conjugate of X₂₉₀

• X₂₃₇ = isotomic conjugate of X₁₈₀₂₄

• X₂₃₇ = complement of X₁₄₉₅₇

• X₂₃₇ = anticomplement of X₂₁₅₃₁

• X₂₃₇ = circumcircle-inverse of X₁₃₁₆

• X₂₃₇ = orthocentroidal-circle-inverse of X₃₇₉₈₈

• X₂₃₇ = orthoptic-circle-of-Steiner-inellipse-inverse of X₃₆₁₈₃

• X₂₃₇ = isogonal conjugate of the anticomplement of X₁₁₆₇₂

• X₂₃₇ = isogonal conjugate of the complement of X₃₉₃₅₅

• X₂₃₇ = isotomic conjugate of the isogonal conjugate of X₉₄₁₈

• X₂₃₇ = isogonal conjugate of the isotomic conjugate of X₅₁₁

• X₂₃₇ = isotomic conjugate of the polar conjugate of X₂₂₁₁

• X₂₃₇ = isogonal conjugate of the polar conjugate of X₂₃₂

• X₂₃₇ = polar conjugate of the isotomic conjugate of X₃₂₈₉

• X₂₃₇ = psi-transform of X₁₅₉₂₀

• X₂₃₇ = X₃₁-complementary conjugate of X₄₀₆₀₁

• X₂₃₇ = X_i-Ceva conjugate of X_j for these (i,j): {2, 40601}, {3, 9475}, {6, 9419}, {98, 6}, {232, 2211}, {511, 3289}, {694, 3051}, {1691, 32748}, {1987, 217}, {2715, 3049}, {4230, 3569}, {6037, 3288}, {9468, 14251}, {14966, 2491}, {17938, 669}, {19189, 232}, {22456, 2451}, {32654, 32}, {34157, 11672}, {44770, 32320}, {46302, 3231}

• X₂₃₇ = X_i-cross conjugate of X_j for these (i,j): {2491, 14966}, {9418, 2211}, {9419, 6}, {11672, 14251}

• X₂₃₇ = cevapoint of X_i and X_j for these (i,j): {3, 3511}, {6, 46272}, {1691, 1971}

• X₂₃₇ = crosspoint of X_i and X_j for these (i,j): {6, 98}, {32, 9468}, {56, 18268}, {232, 511}, {250, 2715}, {325, 34138}, {19189, 41270}, {32654, 34157}

• X₂₃₇ = crosssum of X_i and X_j for these (i,j): {2, 511}, {8, 3948}, {76, 3978}, {98, 287}, {125, 2799}, {264, 16089}, {525, 3150}, {1976, 11610}, {2395, 15630}, {3766, 34387}, {18018, 34237}, {34239, 41194}, {34240, 41195}, {44780, 44781}

• X₂₃₇ = trilinear pole of line {2491, 9419}

• X₂₃₇ = crossdifference of every pair of points on line {2, 647}

• X₂₃₇ = X₃₂-Hirst inverse of X₁₈₄

• X₂₃₇ = X₃-line conjugate of X₂

• X₂₃₇ = X₅₅-beth conjugate of X₂₃₇

• X₂₃₇ = crosspoint of X₃ and X₃₅₁₁ wrt excentral triangle

• X₂₃₇ = crosspoint of X₃ and X₃₅₁₁ wrt tangential triangle

• X₂₃₇ = X₉₂-isoconjugate of X₂₈₇

• X₂₃₇ = trilinear pole of PU(89)

• X₂₃₇ = X_i-isoconjugate of X_j for these (i,j): {1, 290}, {2, 1821}, {4, 336}, {6, 46273}, {31, 18024}, {63, 16081}, {69, 36120}, {75, 98}, {76, 1910}, {85, 15628}, {91, 31635}, {92, 287}, {158, 6394}, {248, 1969}, {264, 293}, {304, 6531}, {308, 3404}, {523, 36036}, {561, 1976}, {656, 22456}, {661, 43187}, {662, 43665}, {685, 14208}, {799, 2395}, {811, 879}, {850, 36084}, {1577, 2966}, {1581, 14382}, {1733, 40428}, {1926, 34238}, {1928, 14601}, {1934, 40820}, {1959, 34536}, {1966, 36897}, {2422, 4602}, {2715, 20948}, {3112, 20021}, {3267, 36104}, {3708, 41174}, {5967, 46277}, {7035, 43920}, {8773, 14265}, {9154, 14210}, {11610, 46244}, {17932, 24006}, {33805, 35906}, {41932, 46238}

• X₂₃₇ = barycentric product X_i*X_j for these {i,j}: {1, 1755}, {3, 232}, {4, 3289}, {5, 41270}, {6, 511}, {25, 36212}, {28, 42702}, {31, 1959}, {32, 325}, {42, 17209}, {48, 240}, {50, 14356}, {55, 43034}, {69, 2211}, {75, 9417}, {76, 9418}, {81, 5360}, {98, 11672}, {99, 2491}, {110, 3569}, {111, 9155}, {112, 684}, {114, 32654}, {184, 297}, {187, 5968}, {206, 34138}, {216, 19189}, {230, 34157}, {248, 2967}, {249, 44114}, {250, 41172}, {290, 9419}, {385, 14251}, {394, 34854}, {512, 2421}, {523, 14966}, {560, 46238}, {577, 6530}, {604, 44694}, {647, 4230}, {648, 39469}, {661, 23997}, {667, 42717}, {669, 2396}, {694, 36213}, {729, 6786}, {868, 23357}, {877, 3049}, {1297, 9475}, {1355, 15628}, {1495, 35910}, {1513, 40799}, {1576, 2799}, {1691, 40810}, {1821, 42075}, {1910, 23996}, {1964, 3405}, {1971, 40804}, {1974, 6393}, {1976, 36790}, {2393, 36823}, {2420, 32112}, {2422, 15631}, {2493, 40083}, {2715, 41167}, {3051, 20022}, {3117, 8840}, {3265, 34859}, {3284, 35908}, {3329, 39684}, {3447, 34349}, {4558, 17994}, {5000, 41197}, {5001, 41196}, {5111, 18873}, {5191, 46787}, {5467, 8430}, {5976, 9468}, {6037, 33569}, {8779, 39265}, {8882, 44716}, {9247, 40703}, {14533, 39569}, {14575, 44132}, {14601, 32458}, {14642, 44704}, {14998, 42743}, {16230, 32661}, {17462, 36051}, {17970, 39931}, {18024, 36425}, {23098, 41932}, {23611, 34536}, {32641, 42751}, {34238, 46888}, {34396, 46807}, {40866, 43112}, {41198, 44779}, {41199, 44778}

• X₂₃₇ = barycentric quotient X_i/X_j for these {i,j}: {1, 46273}, {2, 18024}, {6, 290}, {25, 16081}, {31, 1821}, {32, 98}, {48, 336}, {110, 43187}, {112, 22456}, {163, 36036}, {184, 287}, {206, 31636}, {232, 264}, {240, 1969}, {250, 41174}, {297, 18022}, {325, 1502}, {511, 76}, {512, 43665}, {560, 1910}, {571, 31635}, {577, 6394}, {669, 2395}, {684, 3267}, {868, 23962}, {1501, 1976}, {1513, 40822}, {1576, 2966}, {1691, 14382}, {1692, 14265}, {1755, 75}, {1923, 3404}, {1959, 561}, {1973, 36120}, {1974, 6531}, {1976, 34536}, {1977, 43920}, {2175, 15628}, {2211, 4}, {2396, 4609}, {2421, 670}, {2491, 523}, {2799, 44173}, {2967, 44132}, {3049, 879}, {3051, 20021}, {3289, 69}, {3405, 18833}, {3569, 850}, {4230, 6331}, {5191, 46786}, {5360, 321}, {5968, 18023}, {5976, 14603}, {6393, 40050}, {6530, 18027}, {6786, 30736}, {8789, 34238}, {9155, 3266}, {9233, 14601}, {9247, 293}, {9407, 35906}, {9417, 1}, {9418, 6}, {9419, 511}, {9420, 23878}, {9426, 2422}, {9427, 15630}, {9468, 36897}, {9475, 30737}, {11672, 325}, {12212, 39685}, {14251, 1916}, {14356, 20573}, {14567, 5967}, {14574, 2715}, {14575, 248}, {14585, 17974}, {14601, 41932}, {14602, 40820}, {14966, 99}, {17209, 310}, {17938, 39291}, {17994, 14618}, {19189, 276}, {19627, 14355}, {20022, 40016}, {20968, 11610}, {23098, 32458}, {23611, 36790}, {23996, 46238}, {23997, 799}, {32654, 40428}, {32661, 17932}, {32740, 9154}, {33875, 36822}, {34138, 40421}, {34157, 8781}, {34396, 46806}, {34854, 2052}, {34859, 107}, {36212, 305}, {36213, 3978}, {36823, 46140}, {39469, 525}, {39684, 42006}, {40373, 14600}, {40601, 14957}, {40810, 18896}, {41172, 339}, {41196, 42812}, {41197, 42811}, {41270, 95}, {42075, 1959}, {42702, 20336}, {42717, 6386}, {43034, 6063}, {43112, 46245}, {44114, 338}, {44132, 44161}, {44694, 28659}, {44716, 28706}, {44778, 41195}, {44779, 41194}, {46238, 1928}, {46272, 39058}

• X₂₃₇ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 3, 14096}, {2, 4, 37988}, {2, 20, 37190}, {2, 22, 7467}, {2, 11328, 37338}, {2, 14957, 21531}, {2, 37184, 3}, {2, 37465, 11328}, {2, 46518, 14957}, {3, 22, 46546}, {3, 23, 37916}, {3, 25, 3148}, {3, 381, 35934}, {3, 3148, 37457}, {3, 6638, 37188}, {3, 6660, 37183}, {3, 11328, 2}, {3, 15329, 45662}, {3, 20854, 6660}, {3, 20960, 27369}, {3, 21512, 6636}, {3, 27369, 11326}, {3, 37465, 37338}, {3, 37914, 23}, {3, 41266, 25}, {3, 44894, 441}, {3, 44895, 15000}, {3, 44896, 44887}, {4, 37114, 3}, {6, 160, 20775}, {23, 11145, 34008}, {23, 11146, 34009}, {23, 35296, 37183}, {23, 35298, 3}, {23, 37183, 6660}, {25, 3135, 418}, {25, 11325, 6620}, {25, 20885, 2}, {25, 41266, 20897}, {25, 41275, 37457}, {32, 184, 34396}, {32, 3117, 3051}, {50, 1576, 23200}, {50, 18374, 1576}, {141, 8266, 22062}, {159, 1609, 40947}, {186, 37937, 468}, {187, 1495, 5191}, {187, 3229, 8623}, {187, 5106, 3231}, {187, 21444, 32518}, {206, 571, 14575}, {216, 1843, 23635}, {297, 1513, 2450}, {297, 2450, 868}, {297, 4230, 15143}, {401, 419, 1316}, {418, 27369, 3148}, {419, 11676, 401}, {441, 468, 44887}, {441, 44887, 15000}, {441, 44894, 852}, {468, 44886, 852}, {851, 46513, 854}, {851, 46524, 14953}, {855, 861, 862}, {1113, 1114, 1316}, {1513, 5112, 868}, {1576, 18374, 9407}, {1576, 39231, 50}, {2450, 5112, 297}, {2454, 2455, 44345}, {2479, 2480, 10684}, {3001, 34990, 22087}, {3053, 15270, 682}, {3117, 41278, 32}, {3129, 3130, 23}, {3131, 3132, 6636}, {3135, 20960, 3148}, {3148, 20897, 25}, {3148, 41275, 3}, {3155, 3156, 22}, {3229, 8623, 3231}, {4230, 21525, 868}, {4230, 37123, 5112}, {5004, 5005, 5999}, {5106, 8623, 3229}, {6660, 20854, 23}, {6660, 37183, 37916}, {6660, 37914, 20854}, {7426, 37461, 45662}, {7437, 46501, 46579}, {7437, 46502, 46501}, {8553, 20987, 157}, {8598, 37927, 11634}, {9407, 23200, 1576}, {11007, 21536, 33314}, {11007, 33314, 46562}, {11063, 19596, 7669}, {11257, 40814, 39906}, {11328, 37184, 14096}, {14096, 37338, 2}, {14953, 37018, 46524}, {15244, 15245, 6636}, {15247, 15248, 25}, {18374, 39231, 23200}, {18773, 18774, 21444}, {20775, 40981, 6}, {20854, 35296, 37916}, {20897, 41275, 3148}, {21032, 21036, 20885}, {21522, 46513, 851}, {21536, 33314, 46561}, {34394, 34395, 14567}, {35222, 41328, 3589}, {35296, 37183, 3}, {35298, 37183, 35296}, {35298, 37914, 37916}, {37184, 37465, 2}, {41378, 41379, 184}, {42667, 42668, 5191}, {42789, 42790, 35474}, {44886, 44890, 468}, {44886, 44896, 44894}, {44894, 44896, 44895}, {46501, 46502, 46578}, {46561, 46562, 33314}, {46564, 46568, 857}, {46578, 46579, 46501}, {46600, 46601, 2}

Notes

As a point on the Euler line, X₂₃₇ has Shinagawa coefficients \((EF + F^{2} + S^{2}, -(E + F)^{2} - S^{2})\).