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X(1), X(2), X(3), X(4), X(6), X(9), X(57), X(223), X(282), X(1073), X(1249), X(3341), X(3342), X(3343), X(3344), X(3349), X(3350), X(3351), X(3352), X(3356), X(14481) excenters Ia, Ib, Ic and more generally the extraversions of all the weak points such as X(9), X(57), etc midpoints of ABC, midpoints of altitudes foci of the inscribed Steiner ellipse Q1, Q2, Q3 vertices of the Thomson triangle, see below points R0, R1, R2, R3 described in the page K755 other points below 

Geometric properties : 

The Thomson cubic is the isogonal pK with pivot the centroid G = X(2). See Table 27. It is sometimes called 17point cubic in older literature. It is the complement of the Lucas cubic. See Table 21 for cubics anharmonically equivalent to the Thomson cubic. The isotomic transform of K002 is K184. The Psi transform of K002 is Q119. The symbolic substitution SS{a > √a} transforms K002 into K363. The Thomson isogonal transform of K002 is the central cubic K758. K002 is a member of the class CL043 : it meets the circumcircle at A, B, C and three other points Q1, Q2, Q3 lying on several rectangular hyperbolas (see below). The tangents at these points are concurrent at the Lemoine point X(5646) of the Thomson triangle, a point lying on the lines X(2)X(1350) and X(64)X(631). See K346 and also Q063. Locus properties :


Related papers 

Points on the Thomson cubic 

The following table gives the repartition of points on K002 : for any P on K002, the points P*, Q= G/P, R, T are its isogonal conjugate, GCeva conjugate, Kcrossconjugate, tangential respectively and all these points lie on K002. Recall that the triples P, P*, X(2)  P, Q, X(6)  P, R, X(3)  Q, G/P*, X(4) are collinear on K002. Construction of T : let N be the harmonic conjugate of G with respect to P and P* and let N* be its isogonal conjugate. The tangents at P and P* pass through N*. T also lies on the line passing through G/P* and (G/P)*. The green points are weak points hence their extraversions lie on K002. When a point is not in ETC, a SEARCH number is given. 



P1, P2, P3, P4, P5, P6 are the tangentials of X(3342), X(3343), X(3344), X(3351), X(3350), X(3352) respectively. These points are rather complicated. 

Group law on the Thomson cubic As in the page K004, a group law is defined on K002 with neutral element X(6). For P, Q on K002, P + Q is the X(2)Ceva conjugate of the third point of K002 on the line PQ. Each center P on K002 is associated with an integer n. Three points are collinear if and only if the sum of the corresponding integers is 4. T is the tangential of P. 





If P and P' are two points with integers n and n' then : • P and P' are X(6)cross conjugates hence collinear with X(3) if and only if n + n' = 0, • P and P' are isogonal conjugates hence collinear with X(2) if and only if n + n' = 2, • P and P' are X(2)Ceva conjugates hence collinear with X(6) if and only if n + n' = 4, When n + n' = 6, the points are collinear with X(4) and when n + n' = 8, the points are collinear with X(1073). This is easily adapted for other sums n + n'. The tangential T of P corresponds to 4  2n hence the X(2)Ceva conjugate of T corresponds to 2n. For example, the point P(9) = X(46985) is the X(6)crossconjugate of X(3351) and the isogonal conjugate of P(11) = X(46984). Note that P is a weak point when n is an odd integer. 

K002 contains :
These points R1, R2, R3 lie on the sidelines of Q1Q2Q3, triangle formed by the intersections of K002 and the circumcircle we shall call the Thomson triangle. More informations below. R1, R2, R3 also lie on :
Obviously, the lines GQi and KRi are parallel. Q1, Q2, Q3 also lie on each rectangular hyperbola of the pencil generated by :
G is the orthocenter of Q1Q2Q3 hence (H1) is the Jerabek hyperbola of Q1Q2Q3. See a generalization in : How pivotal cubics intersect the circumcircle The tangents at these points Q1, Q2, Q3 concur at X on the lines X(2)X(1350) and X(64)X(631). X is the Lemoine point of the Thomson triangle Q1Q2Q3, now X(5646) in ETC. The two triangles Q1Q2Q3 and R1R2R3 are perspective at Y on the lines X(6)X(376) and X(39)X(631) with first coordinate : 3a^2(a^2+4b^2+4c^2)+(b^2c^2)^2 and SEARCH number : 1.3852076515. Y is now X(14482) in ETC. 

Thomson cubic and Steiner inellipse 

The tangents at R1, R2, R3 to K002 concur at H. 

The Thomson triangle and the ThomsonJerabek hyperbola 

Recall that we call Thomson triangle the triangle T whose vertices Q1, Q2, Q3 are the three (always real) points where the Thomson cubic meets the circumcircle of ABC again. This triangle is always acutangle, see a proof in §1.6 here. See Table 56 for related curves. Here is a list of properties of this triangle, some of them already mentioned above.
More informations here. K346 is the locus of the poles of pKs meeting the circumcircle at A, B, C, Q1, Q2, Q3. This is the case of K167 = pK(X184, X6) and K172 = pK(X32, X3). The locus of the pivots of such cubics is K002 itself and the locus of the isopivots is K172. *** Three simple hyperbolas (HA), (HB), (HC) are hyperbolas passing through the vertices of the Thomson triangle. 

(HA) is the Thomson isogonal transform of the sideline BC. (HB), (HC) are defined likewise. (HA) passes through the reflection A' of A about O with tangent passing through Ka, the Avertex of the circumcevian triangle of K. If AA' is trisected by points A" and Oa, then A" lies on (HA) and Oa is the center of (HA). The asymptotes of (HA) are perpendicular to the sidelines AB, AC. The axes of (HA) are parallel at Oa to the bisectors at A of ABC. Two points on BC, harmonically conjugated in B and C, have their Thomson isogonal conjugates symmetric in Oa. Two points on BC, symmetric about the midpoint of BC, have their Thomson isogonal conjugates on a line parallel to A'Ka. 

Let P be a point with cevian triangle PaPbPc. The trilinear polar of the isogonal conjugate of P meets BC at Qa. With P = p : q : r, we have Pa = 0 : q : r and Qa = 0 : b^2 r :  c^2 q. Note that the barycentric product Pa × Qa is the center of the AApollonius circle. Pa and Qa have their Thomson isogonal conjugates on a perpendicular to BC. A2 and A1 are obtained when P = X(1). α' and A' are obtained when P = X(6). A" and α" are obtained when P = X(2).


The following list shows several centers of the Thomson Triangle and their counterparts in ABC (Chris van Tienhoven, Peter Moses). For example, {1,5373} means that X(1) in Thomson Triangle is X(5373) in ABC. Those in red remain unchanged, those in blue are swapped. {1,5373}, {2,3524}, {3,3}, {4,2}, {5,549}, {6,5646}, {20,376}, {24,1995}, {30,30}, {54,5888}, {64,6}, {74,110}, {107,98}, {110,74}, {122,2482}, {125,5642}, {133,6055}, {185,5650}, {186,23}, {265,5655}, {381,5054}, {382,381}, {399,10620}, {403,7426}, {459,5304}, {476,477}, {477,476}, {520,512}, {523,523}, {526,526}, {546,140}, {550,8703}, {924,8675}, {1073,5024}, {1075,6194}, {1113,1113}, {1114,1114}, {1147,4550}, {1204,5651}, {1294,99}, {1300,1302}, {1301,111}, {1304,842}, {1498,1350}, {1657,3534}, {2071,7464}, {2574,2575}, {2575,2574}, {2693,691}, {2777,542}, {2972,9155}, {3090,3523}, {3091,631}, {3146,4}, {3183,9740}, {3346,9741}, {3357,182}, {3426,5544}, {3515,25}, {3516,7484}, {3517,5020}, {3520,7496}, {3529,20}, {3532,154}, {3543,3545}, {3627,5}, {3628,3530}, {3830,5055}, {3853,547}, {4240,7422}, {5073,3830}, {5076,1656}, {5663,5663}, {5895,599}, {5896,112}, {5897,1296}, {6000,511}, {6080,2698}, {6086,804}, {6241,7998}, {6247,597}, {6526,7735}, {6622,4232}, {6759,3098}, {7687,5972}, {7689,8717}, {8057,1499}, {8798,39}, {9033,690} *** If M is a point with isogonal conjugate M* with respect to ABC then the isogonal conjugate of M with respect to T is M_{T}*, the centroid of the antipedal triangle of M* in ABC (Randy Hutson). The following list gives pairs of such points {M, M_{T}*}, (Peter Moses, updated 20230728). For example, {1,165} means that the isogonal conjugate of X(1) with respect to the Thomson Triangle is X(165) in ABC. More informations here. {1,165}, {2,3}, {4,154}, {5,6030}, {6,376}, {9,3576}, {13,34317}, {14,34318}, {15,5463}, {16,5464}, {20,3167}, {22,5654}, {23,5655}, {25,5656}, {30,110}, {35,5659}, {36,5660}, {40,3158}, {54,35885}, {55,5657}, {56,5658}, {57,52026}, {61,49901}, {62,49902}, {64,42452}, {74,523}, {84,34499}, {98,512}, {99,511}, {100,517}, {101,516}, {102,522}, {103,514}, {104,513}, {105,3309}, {106,3667}, {107,6000}, {108,6001}, {109,515}, {111,1499}, {112,1503}, {186,15131}, {187,6054}, {198,5603}, {223,52027}, {262,47052}, {354,3651}, {371,13712}, {372,13835}, {381,7712}, {392,4220}, {476,5663}, {477,526}, {518,1292}, {519,1293}, {520,1294}, {521,1295}, {524,1296}, {525,1297}, {527,28291}, {528,2742}, {529,39635}, {530,9202}, {531,9203}, {532,39636}, {533,39637}, {535,39638}, {536,28474}, {537,28520}, {538,39639}, {539,20185}, {541,9060}, {542,691}, {543,2709}, {544,39640}, {545,28293}, {549,5888}, {550,55038}, {551,37508}, {573,47040}, {574,22712}, {674,44876}, {688,53889}, {689,55005}, {690,842}, {698,30254}, {699,30217}, {726,28469}, {727,28470}, {729,32472}, {736,39629}, {739,28475}, {740,6010}, {741,6002}, {752,28467}, {753,28468}, {754,53885}, {755,32473}, {758,6011}, {759,6003}, {789,55004}, {804,2698}, {805,2782}, {812,12032}, {813,28850}, {814,29009}, {815,29010}, {824,28844}, {825,28845}, {826,29011}, {827,29012}, {830,53892}, {834,45136}, {840,2826}, {841,9003}, {843,2793}, {891,29348}, {898,29349}, {900,953}, {901,952}, {902,24808}, {907,29181}, {912,13397}, {915,15313}, {916,1305}, {917,8676}, {918,28838}, {924,1300}, {925,13754}, {926,2724}, {927,2808}, {928,2723}, {929,2807}, {930,1154}, {932,15310}, {933,18400}, {934,971}, {935,2781}, {972,3900}, {991,47039}, {1113,2575}, {1114,2574}, {1137,11228}, {1141,1510}, {1249,10606}, {1289,34146}, {1290,2771}, {1291,32423}, {1301,15311}, {1302,14915}, {1303,32428}, {1304,2777}, {1308,2801}, {1309,2818}, {1350,9741}, {1379,3414}, {1380,3413}, {1381,3308}, {1382,3307}, {1670,33707}, {1671,33708}, {2080,8592}, {2222,2800}, {2291,28292}, {2370,32475}, {2373,30209}, {2374,20186}, {2378,27551}, {2379,27550}, {2382,28521}, {2383,20184}, {2384,28294}, {2390,32704}, {2393,30247}, {2687,8674}, {2688,2774}, {2689,2779}, {2690,2772}, {2691,2836}, {2692,2842}, {2693,9033}, {2694,2850}, {2695,2773}, {2696,2854}, {2697,9517}, {2699,2787}, {2700,2786}, {2701,2792}, {2702,2784}, {2703,2783}, {2704,2795}, {2705,2796}, {2706,2797}, {2707,2798}, {2708,2785}, {2710,2799}, {2711,2788}, {2712,2789}, {2713,2790}, {2714,2791}, {2715,2794}, {2716,3738}, {2717,3887}, {2718,2827}, {2719,2828}, {2720,2829}, {2721,2830}, {2722,2831}, {2725,2820}, {2726,2821}, {2727,2822}, {2728,2823}, {2729,2824}, {2730,2835}, {2731,2841}, {2732,2846}, {2733,2849}, {2734,8677}, {2735,2852}, {2736,2809}, {2737,2810}, {2738,2811}, {2739,2812}, {2740,2813}, {2741,9518}, {2743,2802}, {2744,2803}, {2745,2804}, {2746,2805}, {2747,2806}, {2748,9519}, {2749,9520}, {2750,9521}, {2751,2814}, {2752,2775}, {2753,9522}, {2754,9523}, {2755,9524}, {2756,9525}, {2757,2815}, {2758,2776}, {2759,9526}, {2760,9527}, {2761,9528}, {2762,2816}, {2763,9529}, {2764,9530}, {2765,2817}, {2766,2778}, {2767,9531}, {2768,2819}, {2769,9532}, {2770,2780}, {2825,53182}, {2848,53914}, {2867,53795}, {2881,53912}, {3098,7757}, {3357,51877}, {3522,5644}, {3524,5646}, {3534,9716}, {3563,3566}, {3564,3565}, {3800,29180}, {3827,26706}, {3849,6233}, {3880,30236}, {3906,14388}, {3907,29056}, {3910,29206}, {4083,15323}, {4108,45723}, {4132,53905}, {4160,53896}, {4588,28160}, {4715,28318}, {4725,28327}, {4777,28159}, {4778,28193}, {4785,53900}, {4802,28145}, {4912,28320}, {4926,28203}, {4962,28235}, {4971,28305}, {4977,28173}, {5092,31168}, {5237,49952}, {5238,49953}, {5373,5373}, {5544,10304}, {5638,6039}, {5639,6040}, {5643,8703}, {5645,15688}, {5648,7464}, {5652,7418}, {5653,36166}, {5731,47041}, {5762,43344}, {5840,6099}, {5841,43345}, {5842,15439}, {5844,28218}, {5846,28480}, {5847,28477}, {5851,53887}, {5853,53888}, {5854,53897}, {5897,8057}, {5951,8702}, {5965,53884}, {5966,32478}, {5969,53893}, {5994,41023}, {5995,41022}, {6004,53899}, {6005,29310}, {6006,28233}, {6013,29311}, {6014,28234}, {6078,28915}, {6079,53790}, {6080,53803}, {6082,33962}, {6083,53794}, {6084,28914}, {6085,44873}, {6086,44874}, {6088,6093}, {6194,32524}, {6200,55041}, {6236,8705}, {6323,8704}, {6362,53910}, {6363,48257}, {6366,53911}, {6368,18401}, {6371,53902}, {6372,29308}, {6396,55040}, {6905,33883}, {6998,35270}, {7422,34291}, {7426,45019}, {7927,29316}, {7950,29322}, {7953,29317}, {7954,29323}, {8058,53915}, {8600,32479}, {8652,28146}, {8673,34168}, {8675,43660}, {8678,45137}, {8679,53898}, {8680,36516}, {8681,20187}, {8686,30198}, {8687,29207}, {8694,28194}, {8696,28321}, {8697,28204}, {8699,28236}, {8700,28306}, {8701,28174}, {8708,29309}, {8760,43363}, {8999,53916}, {9000,53908}, {9001,53907}, {9002,53904}, {9004,53901}, {9007,53909}, {9013,53903}, {9019,44061}, {9023,48260}, {9025,39631}, {9027,30256}, {9028,53906}, {9031,53913}, {9051,53917}, {9084,30230}, {9479,53894}, {9756,51880}, {9830,13241}, {9909,32605}, {10420,17702}, {10423,36201}, {10425,23698}, {10628,52998}, {10645,50859}, {10646,50860}, {11636,11645}, {11676,33876}, {12073,53890}, {12074,19924}, {12092,30522}, {12243,33999}, {13152,33643}, {13391,20189}, {13398,44665}, {13597,20188}, {13858,34313}, {13859,34314}, {13860,34099}, {14074,15726}, {14077,15731}, {14482,31884}, {14538,36775}, {14839,53891}, {14924,19708}, {14979,45147}, {14984,53895}, {14988,33637}, {15055,31945}, {15312,53886}, {15728,30199}, {15733,30237}, {15907,25149}, {16168,16170}, {16169,16171}, {16806,44666}, {16807,44667}, {17132,28295}, {17133,28307}, {17222,28296}, {17223,28308}, {17764,28489}, {17765,28575}, {17766,28486}, {17769,28506}, {17770,28483}, {17771,28578}, {17772,28492}, {20219,38454}, {20403,53605}, {20404,53793}, {21158,38412}, {23699,35188}, {23875,29042}, {23876,29045}, {25150,54049}, {25424,32515}, {28147,28149}, {28148,28150}, {28151,28153}, {28152,28154}, {28155,28157}, {28156,28158}, {28161,28163}, {28162,28164}, {28165,28167}, {28166,28168}, {28169,28171}, {28170,28172}, {28175,28177}, {28176,28178}, {28179,28181}, {28180,28182}, {28183,28185}, {28184,28186}, {28187,28189}, {28188,28190}, {28195,28197}, {28196,28198}, {28199,28201}, {28200,28202}, {28205,28207}, {28206,28208}, {28209,28211}, {28210,28212}, {28213,28215}, {28214,28216}, {28217,28219}, {28221,28223}, {28222,28224}, {28225,28227}, {28226,28228}, {28229,28231}, {28230,28232}, {28297,28299}, {28298,28300}, {28301,28303}, {28302,28304}, {28309,28311}, {28310,28312}, {28313,28315}, {28314,28316}, {28317,28319}, {28322,28324}, {28323,28325}, {28326,28328}, {28329,28331}, {28330,28332}, {28333,28335}, {28334,28336}, {28337,28339}, {28338,28340}, {28471,28473}, {28476,28478}, {28479,28481}, {28485,28487}, {28488,28490}, {28491,28493}, {28494,28496}, {28495,28497}, {28498,28500}, {28499,28501}, {28502,28504}, {28505,28507}, {28508,28510}, {28509,28511}, {28512,28514}, {28513,28515}, {28516,28518}, {28517,28519}, {28522,28524}, {28523,28525}, {28526,28528}, {28527,28529}, {28530,28532}, {28531,28533}, {28534,28536}, {28535,28537}, {28538,28540}, {28539,28541}, {28542,28544}, {28543,28545}, {28546,28548}, {28547,28549}, {28550,28552}, {28551,28553}, {28558,28560}, {28559,28561}, {28562,28564}, {28563,28565}, {28566,28568}, {28567,28569}, {28570,28572}, {28571,28573}, {28574,28576}, {28577,28579}, {28582,28584}, {28583,28585}, {28840,28842}, {28846,28848}, {28847,28849}, {28851,28853}, {28852,28854}, {28855,28857}, {28856,28858}, {28859,28861}, {28863,28865}, {28864,28866}, {28867,28869}, {28868,28870}, {28871,28873}, {28872,28874}, {28875,28877}, {28878,28880}, {28879,28881}, {28882,28884}, {28883,28885}, {28886,28888}, {28887,28889}, {28890,28892}, {28891,28893}, {28894,28896}, {28895,28897}, {28898,28900}, {28899,28901}, {28902,28904}, {28903,28905}, {28906,28908}, {28907,28909}, {28910,28912}, {28911,28913}, {29013,29015}, {29014,29016}, {29017,29019}, {29018,29020}, {29021,29023}, {29022,29024}, {29025,29027}, {29026,29028}, {29029,29031}, {29030,29032}, {29033,29035}, {29034,29036}, {29037,29039}, {29038,29040}, {29041,29043}, {29044,29046}, {29047,29049}, {29048,29050}, {29051,29053}, {29052,29054}, {29055,29057}, {29058,29060}, {29059,29061}, {29062,29064}, {29063,29065}, {29066,29068}, {29067,29069}, {29070,29072}, {29071,29073}, {29074,29076}, {29075,29077}, {29078,29080}, {29079,29081}, {29082,29084}, {29083,29085}, {29086,29088}, {29087,29089}, {29090,29092}, {29091,29093}, {29094,29096}, {29095,29097}, {29098,29100}, {29099,29101}, {29102,29104}, {29103,29105}, {29106,29108}, {29107,29109}, {29110,29112}, {29111,29113}, {29208,29210}, {29209,29211}, {29212,29214}, {29213,29215}, {29216,29218}, {29217,29219}, {29220,29222}, {29221,29223}, {29226,29228}, {29227,29229}, {29232,29234}, {29233,29235}, {29240,29242}, {29241,29243}, {29252,29254}, {29253,29255}, {29256,29258}, {29257,29259}, {29260,29262}, {29261,29263}, {29280,29282}, {29281,29283}, {29284,29286}, {29285,29287}, {29288,29290}, {29289,29291}, {29294,29296}, {29295,29297}, {29298,29300}, {29299,29301}, {29304,29306}, {29305,29307}, {29312,29314}, {29313,29315}, {29318,29320}, {29319,29321}, {29324,29326}, {29325,29327}, {29328,29330}, {29329,29331}, {29332,29334}, {29333,29335}, {29336,29338}, {29337,29339}, {29340,29342}, {29341,29343}, {29344,29346}, {29345,29347}, {29350,29352}, {29351,29353}, {29354,29356}, {29358,29360}, {29362,29364}, {29363,29365}, {29366,29368}, {29367,29369}, {29370,29372}, {29371,29373}, {30210,39431}, {30211,39434}, {30212,39435}, {30213,39436}, {32228,53938}, {33639,50708}, {33646,34474}, {33705,33706}, {33850,50967}, {33927,47050}, {38455,39628}, {39162,39163}, {39164,39165}, {39469,48259}, {40989,52053}, {40990,52054}, {41376,53015}, {44059,44670}, {44062,45780}, {47044,47049}, {47045,47051}, {47046,47047}, {53606,53792}

