X₆₉₀

CROSSDIFFERENCE OF LINE X₆ AND X₁₁₀

As the isogonal conjugate of a point on the circumcircle, X₆₉₀ lies on the line at infinity.

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

X₆₉₀ (CROSSDIFFERENCE OF LINE X₆ AND X₁₁₀) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {74, 98}, {99, 110}

• Related Centers: X₆₉₁ (TRILINEAR POLE OF LINE X₆X₁₁₀), X₇₄ (ISOGONAL CONJUGATE OF EULER INFINITY POINT), X₉₈ (TARRY POINT), X₉₉ (STEINER POINT), X₁₁₀ (FOCUS OF KIEPERT PARABOLA)

Trilinears: \(bc(b^{2} - c^{2})(2a^{2} - b^{2} - c^{2}) : :\)

Barycentrics: \((\cot B - \cot C) (2 \cot A - \cot B - \cot C) : :\)

▸ Incident Central Lines376 lines

{2, 5465}, {3, 6334}, {4, 11005}, {5, 39509}, {6, 36880}, {10, 18004}, {13, 16255}, {14, 16256}, {30, 511}, {39, 2491}, {67, 10097}, {69, 13232}, {74, 98}, {76, 14295}, {83, 18010}, {99, 110}, {111, 39446}, {113, 114}, {115, 125}, {146, 147}, {148, 3448}, {182, 39499}, {226, 18006}, {265, 6321}, {321, 18003}, {351, 1641}, {399, 13188}, {599, 34206}, {620, 5113}, {669, 2525}, {671, 5466}, {684, 14981}, {685, 16077}, {691, 20404}, {850, 8599}, {882, 1916}, {887, 14824}, {895, 2408}, {905, 42653}, {1112, 5186}, {1177, 39840}, {1281, 44433}, {1495, 32120}, {1511, 33813}, {1539, 22505}, {1569, 9419}, {1962, 14413}, {2254, 2292}, {2378, 11612}, {2379, 11613}, {2395, 34246}, {2422, 14901}, {2435, 11744}, {2485, 7631}, {2489, 7651}, {2492, 14279}, {2510, 8574}, {2528, 8664}, {2530, 42661}, {2533, 7265}, {2650, 4895}, {2682, 5099}, {2931, 39828}, {2935, 39841}, {2936, 34519}, {2948, 13174}, {3005, 17436}, {3023, 3024}, {3027, 3028}, {3029, 3031}, {3044, 3047}, {3095, 45911}, {3111, 9828}, {3265, 8651}, {3426, 14220}, {3455, 42659}, {3743, 3960}, {3762, 4647}, {3801, 4170}, {3806, 8665}, {4010, 4707}, {4027, 13193}, {4049, 11599}, {4080, 18011}, {4088, 4730}, {4122, 4761}, {4444, 18009}, {4729, 4808}, {4736, 13277}, {4773, 8659}, {4846, 15453}, {4983, 21124}, {5026, 6593}, {5092, 14271}, {5095, 5477}, {5108, 9129}, {5181, 21905}, {5461, 9183}, {5467, 14559}, {5476, 11622}, {5485, 18012}, {5489, 9409}, {5503, 43674}, {5655, 8724}, {5916, 5917}, {5976, 38650}, {5987, 14318}, {5989, 38661}, {6027, 44051}, {6033, 7728}, {6036, 6699}, {6054, 10706}, {6055, 11656}, {6130, 11623}, {6132, 6140}, {6226, 7726}, {6227, 7725}, {6233, 6236}, {6319, 7732}, {6320, 7733}, {6323, 6325}, {6545, 30592}, {6721, 12900}, {6722, 6723}, {6771, 16233}, {6774, 16234}, {7472, 9181}, {7709, 15920}, {7731, 39837}, {7779, 13186}, {7893, 13237}, {7970, 7978}, {7983, 7984}, {8030, 9135}, {8591, 9131}, {8593, 41720}, {8663, 16892}, {8782, 13210}, {8980, 8994}, {8997, 8998}, {9080, 9187}, {9115, 30455}, {9117, 30454}, {9161, 43654}, {9178, 34898}, {9209, 22264}, {9507, 11725}, {9759, 9877}, {9860, 9904}, {9861, 9919}, {9862, 9984}, {9864, 12368}, {9880, 44203}, {9966, 16510}, {9970, 12177}, {10053, 10065}, {10069, 10081}, {10086, 10088}, {10089, 10091}, {10099, 10693}, {10113, 22515}, {10117, 32122}, {10118, 39851}, {10119, 39850}, {10168, 11621}, {10190, 36521}, {10264, 15535}, {10278, 36523}, {10293, 14380}, {10620, 12188}, {10663, 39829}, {10664, 39830}, {10681, 39858}, {10682, 39859}, {10721, 10722}, {10723, 10733}, {10752, 10753}, {10767, 10768}, {10769, 10778}, {10992, 30714}, {11061, 25052}, {11161, 13169}, {11257, 38520}, {11472, 15928}, {11579, 21732}, {11608, 18013}, {11611, 18015}, {11616, 12584}, {11632, 14849}, {11709, 11710}, {11711, 11720}, {11723, 11724}, {12041, 12042}, {12075, 33294}, {12121, 38730}, {12131, 12133}, {12168, 39803}, {12176, 12192}, {12178, 12327}, {12179, 12365}, {12180, 12366}, {12181, 12369}, {12182, 12371}, {12183, 12372}, {12184, 12373}, {12185, 12374}, {12186, 12377}, {12187, 12378}, {12189, 12381}, {12190, 12382}, {12228, 39805}, {12236, 39806}, {12258, 21181}, {12273, 39807}, {12284, 39808}, {12295, 39809}, {12302, 39812}, {12310, 13175}, {12319, 39813}, {12375, 35878}, {12376, 35879}, {12383, 13172}, {12494, 34113}, {12596, 39819}, {12661, 39821}, {12888, 39822}, {12891, 39823}, {12892, 39824}, {12893, 39825}, {12901, 39831}, {12902, 38733}, {12903, 13182}, {12904, 13183}, {13171, 39832}, {13173, 13204}, {13176, 13208}, {13177, 13209}, {13178, 13211}, {13179, 13212}, {13180, 13213}, {13181, 13214}, {13184, 13215}, {13185, 13216}, {13187, 14316}, {13189, 13217}, {13190, 13218}, {13198, 39834}, {13201, 39836}, {13202, 39838}, {13203, 39842}, {13233, 32305}, {13234, 45163}, {13248, 39848}, {13287, 39852}, {13288, 39853}, {13289, 39854}, {13293, 39860}, {13417, 39846}, {13642, 13643}, {13653, 13654}, {13761, 13762}, {13773, 13774}, {13967, 13969}, {13989, 13990}, {14061, 15059}, {14094, 23235}, {14337, 14339}, {14338, 14340}, {14345, 15131}, {14430, 21020}, {14606, 34364}, {14639, 14644}, {14643, 14850}, {14653, 16508}, {14670, 23108}, {14683, 19598}, {15035, 21166}, {15054, 38664}, {15055, 34473}, {15061, 38224}, {15070, 15099}, {15088, 15092}, {15303, 18311}, {15328, 34802}, {16003, 16220}, {16093, 32244}, {16111, 18556}, {16163, 38738}, {17708, 32729}, {17838, 39820}, {17847, 39849}, {18008, 40016}, {18309, 25562}, {18310, 19662}, {18932, 39804}, {18947, 39833}, {19055, 19059}, {19056, 19060}, {19108, 19110}, {19109, 19111}, {19138, 39811}, {19193, 39814}, {19208, 39843}, {19456, 39810}, {19469, 39815}, {19479, 39818}, {19482, 39826}, {19483, 39827}, {19504, 39839}, {19505, 39844}, {19506, 39847}, {19507, 39855}, {19508, 39856}, {20127, 38741}, {20397, 20398}, {21649, 39817}, {22089, 34952}, {22159, 37085}, {22504, 22583}, {22514, 22586}, {23306, 39816}, {23315, 39845}, {23514, 23515}, {23992, 41177}, {24978, 36253}, {24981, 30220}, {25315, 25320}, {25322, 25328}, {25325, 25329}, {25334, 25335}, {25556, 32135}, {31290, 40502}, {32111, 32112}, {32271, 33752}, {34127, 34128}, {34453, 34454}, {34801, 40048}, {34964, 39512}, {35824, 35826}, {35825, 35827}, {36165, 45284}, {36518, 36519}, {37853, 38747}, {38481, 38482}, {38497, 38498}, {38499, 38508}, {38555, 38556}, {38557, 38566}, {38626, 38627}, {38628, 38632}, {38633, 38634}, {38635, 38638}, {38641, 38642}, {38653, 38654}, {38723, 38731}, {38724, 38732}, {38725, 38735}, {38726, 38736}, {38727, 38737}, {38728, 38739}, {38729, 38740}, {38742, 38788}, {38743, 38789}, {38744, 38790}, {38745, 38791}, {38746, 38792}, {38748, 38793}, {38750, 38794}, {38751, 38795}, {39482, 39491}, {39495, 39501}, {39513, 39518}, {40486, 40513}, {41189, 41193}, {41190, 41192}, {43532, 43665}

▸ Functional Relations29 relations

• X₆₉₀ = isogonal conjugate of X₆₉₁

• X₆₉₀ = isotomic conjugate of X₈₉₂

• X₆₉₀ = X₆₇-Ceva conjugate of X₁₂₅

• X₆₉₀ = crosssum of X_i and X_j for these (i,j): (6, 351), (187, 512), (523, 858)

• X₆₉₀ = crossdifference of every pair of points on line X₆X₁₁₀

• X₆₉₀ = orthopoint of X₅₄₂

• X₆₉₀ = X₂-Ceva conjugate of X₂₃₉₉₂

• X₆₉₀ = polar conjugate of the isotomic conjugate of X₁₄₄₁₇

• X₆₉₀ = X₅₂₃-of-1st-Brocard-triangle

• X₆₉₀ = X₅₁₂-of-4th-Brocard-triangle

• X₆₉₀ = X₅₁₂-of-orthocentroidal-triangle

• X₆₉₀ = X₅₁₂-of-X₄-Brocard-triangle

• X₆₉₀ = X₁₄₉₉-of-McCay-triangle

• X₆₉₀ = X₁₄₉₉-of-anti-McCay-triangle

• X₆₉₀ = crosspoint of X(I) and X(J) for these (I,J): (99, 671), (110, 1177)

• X₆₉₀ = intersection of tangents to Steiner inellipse at X₁₁₅ and X₂₄₈₂

• X₆₉₀ = crosspoint wrt medial triangle of X₁₁₅ and X₂₄₈₂

• X₆₉₀ = trilinear pole of line X₁₆₄₈X₁₆₄₉

• X₆₉₀ = barycentric product X₅₂₃*X₅₂₄

• X₆₉₀ = 1st-Brocard-isogonal conjugate of X₁₃₁₆

• X₆₉₀ = 1st-Brocard-isotomic conjugate of X₆

• X₆₉₀ = isogonal conjugate of the anticomplement of X₅₀₉₉

• X₆₉₀ = isotomic conjugate of the anticomplement of X₂₃₉₉₂

• X₆₉₀ = isotomic conjugate of the complement of X₃₉₃₅₆

• X₆₉₀ = isotomic conjugate of the isogonal conjugate of X₃₅₁

• X₆₉₀ = isogonal conjugate of the isotomic conjugate of X₃₅₅₂₂

• X₆₉₀ = isotomic conjugate of the polar conjugate of X₁₄₂₇₃

• X₆₉₀ = polar conjugate of the isotomic conjugate of X₁₄₄₁₇

• X₆₉₀ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 9144, 5465}, {2, 9185, 9189}, {3, 14270, 39477}, {4, 18331, 11005}, {4, 44427, 16230}, {10, 22037, 18004}, {74, 22265, 98}, {99, 15342, 110}, {115, 125, 15359}, {115, 15357, 125}, {125, 1637, 42736}, {125, 16278, 115}, {351, 1649, 9125}, {351, 11183, 45680}, {351, 14424, 14417}, {671, 5466, 18007}, {1637, 9148, 45688}, {1640, 3569, 1637}, {2254, 2292, 42666}, {2489, 7652, 7651}, {2682, 35582, 14114}, {3268, 9123, 9168}, {3268, 9147, 45687}, {3268, 39904, 5652}, {3569, 9148, 11182}, {4120, 30574, 14431}, {4750, 14432, 14419}, {5099, 38395, 2682}, {5113, 45693, 11176}, {5652, 9147, 5027}, {5653, 9138, 9208}, {6132, 11615, 6140}, {6321, 15545, 265}, {6699, 33511, 6036}, {7472, 14999, 9181}, {8552, 11615, 6132}, {9123, 9168, 45687}, {9125, 14417, 1649}, {9138, 14698, 14697}, {9138, 14932, 9185}, {9144, 11006, 2}, {9147, 9168, 9123}, {9148, 14420, 1637}, {9148, 36255, 125}, {9180, 14932, 5465}, {9185, 9191, 2}, {9185, 14698, 14932}, {9185, 14932, 14697}, {9194, 9195, 9189}, {9200, 9201, 1637}, {9200, 14447, 9201}, {9201, 14446, 9200}, {9204, 9205, 14417}, {9485, 44010, 9131}, {11176, 24284, 45693}, {13169, 14833, 11161}, {13291, 36255, 1637}, {13636, 13722, 8371}, {14270, 44826, 3}, {14419, 30595, 4750}, {14419, 30605, 14432}, {14420, 42734, 9200}, {14420, 42735, 9201}, {14420, 42737, 13291}, {14446, 14447, 1637}, {14643, 14850, 15561}, {14698, 39905, 5653}, {15357, 16278, 15359}, {30508, 30509, 9168}, {30595, 30605, 14419}, {31953, 42738, 1637}, {32193, 45689, 44564}, {32312, 32313, 351}, {42733, 42738, 115}, {42734, 42735, 9148}, {45327, 45336, 44564}, {45336, 45689, 45692}

Notes

Let N_aN_bN_c and N_a'N_b'N_c' be the outer and inner Napoleon triangles, respectively. Let A' be the isogonal conjugate of N_a wrt N_a'N_b'N_c', and define B' and C' cyclically. Let A″ be the isogonal conjugate of N_a' wrt N_aN_bN_c, and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X₆₉₀. (Randy Hutson, November 30, 2015)