X₂₃₂

X₂-CEVA CONJUGATE OF X₁₃₂

X₂₃₂ = polar conjugate of the isogonal conjugate of X₂₃₇

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

X₂₃₂ (X₂-CEVA CONJUGATE OF X₁₃₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 25}, {4, 39}

• Related Centers: X₂ (CENTROID), X₁₃₂ (∩X₄X₃₂), X₆ (SYMMEDIAN POINT), X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES), X₄ (ORTHOCENTER), X₃₉ (BROCARD MIDPOINT)

Trilinears: \(\tan A \cos(A + \omega ) : \tan B \cos(B + \omega ) : \tan C \cos(C + \omega )\)

Barycentrics: \(\sin A \tan A \cos(A + \omega ) : \sin B \tan B \cos(B + \omega ) : \sin C \tan C \cos(C + \omega )\)

Search Value (6, 9, 13): \(0.669458175105\)

▸ Alternate Coordinates1 form

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

X₂₃₂ lies on the cubics K222, K493, K756, K781, K782, K783, K785, K787, K1097.

▸ Incident Central Lines268 lines

{2, 216}, {3, 1968}, {4, 39}, {5, 27376}, {6, 25}, {19, 444}, {20, 22401}, {22, 577}, {23, 250}, {24, 32}, {26, 10316}, {30, 14961}, {33, 2276}, {34, 2275}, {50, 3447}, {53, 427}, {69, 37187}, {98, 41204}, {99, 15014}, {111, 1304}, {112, 186}, {114, 38970}, {115, 403}, {132, 1513}, {136, 38975}, {183, 9308}, {185, 32445}, {211, 15897}, {217, 389}, {230, 231}, {233, 37990}, {235, 5254}, {237, 9475}, {240, 42702}, {248, 32649}, {251, 8882}, {297, 325}, {316, 40889}, {317, 7774}, {340, 7779}, {378, 574}, {385, 648}, {406, 5283}, {419, 6531}, {420, 3229}, {428, 5421}, {451, 16589}, {458, 11174}, {511, 2211}, {538, 41676}, {543, 37855}, {566, 5094}, {571, 1485}, {607, 26377}, {685, 41932}, {800, 1196}, {1015, 1870}, {1033, 1611}, {1107, 46878}, {1113, 15167}, {1114, 15166}, {1172, 2092}, {1180, 3087}, {1184, 8573}, {1216, 41480}, {1235, 3934}, {1297, 32687}, {1368, 42459}, {1384, 11181}, {1500, 6198}, {1506, 1594}, {1560, 3258}, {1575, 1861}, {1585, 8962}, {1593, 5013}, {1595, 31406}, {1596, 15048}, {1597, 5024}, {1598, 9605}, {1609, 3162}, {1625, 13754}, {1627, 41758}, {1691, 34130}, {1692, 2065}, {1783, 5291}, {1785, 13006}, {1865, 37362}, {1875, 43039}, {1906, 9607}, {1907, 9606}, {1970, 13367}, {1986, 46301}, {1989, 8791}, {1995, 5158}, {2023, 12131}, {2070, 10317}, {2071, 40349}, {2079, 37917}, {2165, 13854}, {2176, 41320}, {2197, 7120}, {2202, 7117}, {2204, 20832}, {2322, 26244}, {2332, 18755}, {2373, 46239}, {2374, 9091}, {2409, 35906}, {2971, 5140}, {3053, 3172}, {3055, 43458}, {3088, 31400}, {3089, 5286}, {3091, 26216}, {3092, 6422}, {3093, 6421}, {3094, 12294}, {3108, 33631}, {3163, 7426}, {3269, 3331}, {3329, 36794}, {3516, 15815}, {3517, 30435}, {3518, 5007}, {3520, 37512}, {3541, 31401}, {3542, 3767}, {3569, 32112}, {3575, 7745}, {3917, 40805}, {4213, 21838}, {4220, 5317}, {4230, 5968}, {4232, 5304}, {4233, 40129}, {4235, 32456}, {5000, 41196}, {5001, 41197}, {5008, 10986}, {5023, 8778}, {5034, 39588}, {5041, 34484}, {5052, 6403}, {5058, 10880}, {5062, 10881}, {5063, 9609}, {5064, 13351}, {5133, 36412}, {5206, 32534}, {5276, 35973}, {5305, 21841}, {5462, 41334}, {5661, 6795}, {5899, 22121}, {5907, 22416}, {5978, 11093}, {5979, 11094}, {6114, 6116}, {6115, 6117}, {6240, 7747}, {6241, 41367}, {6529, 41368}, {6623, 43448}, {6636, 22052}, {6683, 37125}, {6749, 10301}, {6759, 39643}, {6781, 10295}, {6998, 8747}, {7071, 31477}, {7387, 23115}, {7413, 8748}, {7418, 35908}, {7467, 11574}, {7484, 36751}, {7485, 10979}, {7505, 7746}, {7517, 22120}, {7576, 7753}, {7577, 7603}, {7713, 9575}, {7714, 13341}, {7737, 18533}, {7749, 10018}, {7755, 41366}, {7756, 18560}, {7772, 10594}, {7786, 37337}, {7868, 11331}, {8430, 17994}, {8585, 14685}, {8588, 35472}, {8589, 35473}, {8750, 17735}, {8752, 17969}, {8753, 15268}, {8770, 41489}, {9229, 40413}, {9596, 11392}, {9599, 11393}, {9698, 15559}, {9753, 41371}, {9909, 15905}, {10151, 35903}, {10184, 11548}, {10282, 14585}, {10539, 23128}, {10540, 22146}, {10605, 41376}, {10632, 16258}, {10633, 16257}, {11063, 37920}, {11325, 40325}, {11380, 34870}, {11381, 38297}, {11398, 16502}, {11403, 22332}, {11456, 39913}, {13357, 27369}, {13481, 37453}, {13509, 14157}, {13860, 33971}, {14002, 15860}, {14356, 34349}, {14537, 18559}, {14865, 31652}, {15262, 37777}, {15340, 25739}, {15484, 18494}, {15513, 21844}, {15515, 35477}, {16068, 17980}, {16868, 39565}, {16968, 36103}, {16990, 44134}, {17966, 32674}, {17984, 19566}, {18907, 37458}, {18993, 45502}, {18994, 45503}, {19189, 37123}, {19220, 44269}, {20850, 38292}, {20960, 27373}, {21525, 34859}, {21843, 35486}, {25985, 37661}, {27377, 41624}, {28710, 32816}, {31455, 37119}, {32697, 35296}, {33630, 38282}, {33828, 40411}, {34481, 46432}, {34818, 39951}, {34990, 44377}, {35007, 44879}, {35764, 45513}, {35765, 45512}, {35907, 36166}, {36414, 36416}, {36415, 36423}, {37121, 44732}, {37197, 44518}, {37942, 43291}, {37969, 39176}, {37984, 44468}, {44438, 44526}

▸ Functional Relations39 relations

• X₂₃₂ = midpoint of X_i and X_j for these {i,j}: {237, 38368}, {3269, 3331}, {41676, 44146}

• X₂₃₂ = isogonal conjugate of X₂₈₇

• X₂₃₂ = complement of X₃₀₇₃₇

• X₂₃₂ = polar-circle-inverse of X₃₄₂₃₅

• X₂₃₂ = Moses-radical-circle-inverse of X₄₆₈

• X₂₃₂ = Moses-Parry-circle inverse of X₆₁₀₃

• X₂₃₂ = polar conjugate of X₂₉₀

• X₂₃₂ = complement of the isotomic conjugate of X₁₂₉₇

• X₂₃₂ = isogonal conjugate of the anticomplement of X₁₅₅₉₅

• X₂₃₂ = isogonal conjugate of the complement of X₄₀₈₆₇

• X₂₃₂ = isotomic conjugate of the isogonal conjugate of X₂₂₁₁

• X₂₃₂ = isogonal conjugate of the isotomic conjugate of X₂₉₇

• X₂₃₂ = isotomic conjugate of the polar conjugate of X₃₄₈₅₄

• X₂₃₂ = isogonal conjugate of the polar conjugate of X₆₅₃₀

• X₂₃₂ = polar conjugate of the isotomic conjugate of X₅₁₁

• X₂₃₂ = polar conjugate of the isogonal conjugate of X₂₃₇

• X₂₃₂ = X_i-complementary conjugate of X_j for these (i,j): {31, 132}, {560, 23976}, {1297, 2887}, {8767, 21243}, {32649, 8062}, {34212, 21253}, {35140, 21235}, {36046, 30476}, {43717, 20305}, {44770, 21259}

• X₂₃₂ = X_i-Ceva conjugate of X_j for these (i,j): {2, 132}, {297, 511}, {685, 512}, {1976, 1968}, {2409, 2881}, {3563, 25}, {4230, 17994}, {6530, 34854}, {16081, 4}, {19189, 237}, {32649, 2485}, {43717, 6}

• X₂₃₂ = X_i-cross conjugate of X_j for these (i,j): {237, 511}, {446, 694}, {2211, 34854}, {9475, 6}, {17994, 4230}, {36425, 27375}, {38368, 523}, {38974, 647}, {41172, 3569}, {44114, 16230}

• X₂₃₂ = cevapoint of X_i and X_j for these (i,j): {237, 2211}, {419, 41204}, {1692, 42671}, {2491, 44114}, {3569, 41172}

• X₂₃₂ = crosspoint of X_i and X_j for these (i,j): {2, 1297}, {4, 16081}, {6, 1987}, {25, 17980}, {250, 32697}, {297, 6530}, {685, 23582}, {1976, 9292}, {23964, 32687}, {41200, 41201}

• X₂₃₂ = crosssum of X_i and X_j for these (i,j): {2, 401}, {3, 3289}, {6, 1503}, {69, 12215}, {248, 17974}, {325, 1975}, {684, 3269}, {15526, 39473}, {41198, 41199}

• X₂₃₂ = trilinear pole of line {3569, 17994}

• X₂₃₂ = crossdifference of every pair of points on line {3, 525}

• X₂₃₂ = orthojoin of X₁₃₂

• X₂₃₂ = X₆-Hirst inverse of X₂₅

• X₂₃₂ = X₂₈₁-beth conjugate of X₂₃₂

• X₂₃₂ = perspector of hyperbola {A, B, C,X₄,X₁₁₂,PU(39)}} (centered at X₁₃₂)

• X₂₃₂ = center of circumconic that is locus of trilinear poles of lines passing through X₁₃₂

• X₂₃₂ = intersection of trilinear polars of X₁₁₂, P(39), and U(39)

• X₂₃₂ = crossdifference of PU(37)

• X₂₃₂ = PU(4)-harmonic conjugate of X₆₄₇

• X₂₃₂ = pole wrt polar circle of trilinear polar of X₂₉₀ (line X₂X₆₄₇)

• X₂₃₂ = X₄₈-isoconjugate (polar conjugate) of X₂₉₀

• X₂₃₂ = X₁₉-isoconjugate of X₆₃₉₄

• X₂₃₂ = X_i-isoconjugate of X_j for these (i,j): {1, 287}, {2, 293}, {3, 1821}, {6, 336}, {19, 6394}, {48, 290}, {63, 98}, {69, 1910}, {75, 248}, {77, 15628}, {92, 17974}, {184, 46273}, {255, 16081}, {304, 1976}, {326, 6531}, {394, 36120}, {525, 36084}, {561, 14600}, {647, 36036}, {656, 2966}, {661, 17932}, {662, 879}, {685, 24018}, {799, 878}, {810, 43187}, {822, 22456}, {1577, 43754}, {1799, 3404}, {1820, 31635}, {1966, 15391}, {2349, 35912}, {2395, 4592}, {2715, 14208}, {3265, 36104}, {4575, 43665}, {8766, 9476}, {9247, 18024}, {14601, 40364}, {20021, 34055}

• X₂₃₂ = barycentric product X_i*X_j for these {i,j}: {1, 240}, {3, 6530}, {4, 511}, {5, 19189}, {6, 297}, {19, 1959}, {25, 325}, {30, 35908}, {31, 40703}, {32, 44132}, {34, 44694}, {54, 39569}, {64, 44704}, {69, 34854}, {76, 2211}, {92, 1755}, {98, 2967}, {99, 17994}, {107, 684}, {110, 16230}, {112, 2799}, {114, 3563}, {132, 1297}, {186, 14356}, {237, 264}, {248, 36426}, {250, 868}, {281, 43034}, {286, 5360}, {324, 41270}, {393, 36212}, {419, 40810}, {468, 5968}, {512, 877}, {523, 4230}, {648, 3569}, {685, 41167}, {694, 39931}, {935, 33752}, {1309, 42751}, {1503, 39265}, {1513, 40801}, {1826, 17209}, {1843, 20022}, {1969, 9417}, {1973, 46238}, {1990, 35910}, {2052, 3289}, {2203, 42703}, {2207, 6393}, {2396, 2489}, {2421, 2501}, {2491, 6331}, {3267, 34859}, {3405, 17442}, {4235, 8430}, {4240, 32112}, {5000, 5001}, {5140, 36892}, {5523, 36823}, {5976, 17980}, {6103, 46787}, {6330, 9475}, {6333, 32713}, {6528, 39469}, {6531, 36790}, {6591, 42717}, {7473, 23350}, {8743, 34138}, {8884, 44716}, {9155, 17983}, {9307, 15143}, {9418, 18022}, {10311, 46807}, {11672, 16081}, {14251, 17984}, {14618, 14966}, {15595, 43717}, {18020, 44114}, {20410, 36884}, {23582, 41172}, {23996, 36120}, {23997, 24006}, {34129, 38652}, {34157, 44145}, {40804, 41204}, {44778, 44781}, {44779, 44780}

• X₂₃₂ = barycentric quotient X_i/X_j for these {i,j}: {1, 336}, {3, 6394}, {4, 290}, {6, 287}, {19, 1821}, {24, 31635}, {25, 98}, {31, 293}, {32, 248}, {92, 46273}, {107, 22456}, {110, 17932}, {112, 2966}, {132, 30737}, {162, 36036}, {184, 17974}, {237, 3}, {240, 75}, {264, 18024}, {297, 76}, {325, 305}, {393, 16081}, {419, 14382}, {460, 14265}, {511, 69}, {512, 879}, {607, 15628}, {648, 43187}, {669, 878}, {684, 3265}, {868, 339}, {877, 670}, {1096, 36120}, {1495, 35912}, {1501, 14600}, {1576, 43754}, {1755, 63}, {1843, 20021}, {1959, 304}, {1973, 1910}, {1974, 1976}, {2207, 6531}, {2211, 6}, {2421, 4563}, {2450, 41009}, {2489, 2395}, {2491, 647}, {2501, 43665}, {2799, 3267}, {2967, 325}, {3289, 394}, {3563, 40428}, {3569, 525}, {4230, 99}, {5000, 42812}, {5001, 42811}, {5140, 36874}, {5360, 72}, {5968, 30786}, {6000, 36893}, {6103, 46786}, {6530, 264}, {6531, 34536}, {8430, 14977}, {8743, 31636}, {8753, 9154}, {9155, 6390}, {9417, 48}, {9418, 184}, {9419, 3289}, {9468, 15391}, {9475, 441}, {10311, 46806}, {11672, 36212}, {14251, 36214}, {14356, 328}, {14581, 35906}, {14966, 4558}, {15143, 1975}, {16230, 850}, {17209, 17206}, {17409, 11610}, {17980, 36897}, {17994, 523}, {19189, 95}, {23582, 41174}, {23997, 4592}, {32112, 34767}, {32676, 36084}, {32696, 41173}, {32713, 685}, {34157, 43705}, {34397, 14355}, {34854, 4}, {34859, 112}, {35908, 1494}, {36212, 3926}, {36213, 12215}, {36426, 44132}, {36790, 6393}, {38368, 3150}, {39265, 35140}, {39469, 520}, {39569, 311}, {39931, 3978}, {40601, 14965}, {40703, 561}, {40810, 40708}, {41167, 6333}, {41172, 15526}, {41270, 97}, {42067, 43920}, {42068, 15630}, {42671, 34156}, {42702, 3998}, {43034, 348}, {43717, 9476}, {44080, 11653}, {44089, 40820}, {44102, 5967}, {44114, 125}, {44132, 1502}, {44162, 14601}, {44467, 37858}, {44694, 3718}, {44704, 14615}, {44778, 32619}, {44779, 32618}, {46238, 40364}, {46522, 36822}

• X₂₃₂ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 22240, 216}, {3, 2207, 1968}, {4, 33885, 33842}, {4, 39575, 39}, {6, 25, 10311}, {6, 1971, 8779}, {24, 8743, 32}, {25, 10311, 10985}, {25, 40938, 1194}, {25, 44089, 44099}, {25, 44090, 44089}, {25, 45141, 6}, {39, 3199, 4}, {39, 33842, 33843}, {53, 3815, 427}, {112, 186, 187}, {112, 8744, 14581}, {186, 8744, 112}, {187, 14581, 112}, {230, 1990, 16318}, {230, 16318, 6103}, {403, 5523, 115}, {468, 1990, 6103}, {468, 14580, 3291}, {468, 16318, 230}, {800, 1196, 7735}, {1249, 6353, 7735}, {1495, 8779, 1971}, {1506, 27371, 1594}, {1513, 6530, 132}, {1990, 2493, 14580}, {1990, 6103, 41358}, {1990, 11062, 3003}, {2493, 3003, 3291}, {2493, 11062, 468}, {2493, 16308, 3003}, {2971, 46522, 5140}, {3003, 14580, 6103}, {3172, 3515, 3053}, {3199, 33842, 33885}, {3199, 33843, 33842}, {3542, 41361, 3767}, {5412, 5413, 1974}, {6995, 37665, 3087}, {8105, 8106, 6103}, {8739, 8740, 44102}, {8778, 15750, 5023}, {10641, 10642, 44091}, {14576, 40938, 10311}, {15355, 22240, 2}, {18533, 41370, 7737}, {19128, 41363, 1692}, {33842, 33843, 4}, {38867, 43717, 248}, {41676, 46511, 44146}, {44089, 44090, 44102}, {44099, 44102, 44089}