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There are "officially" two Brocard triangles. Since the isogonal conjugates of their vertices are often present, we shall call them also Brocard triangles. Ω1 and Ω2 are the Brocard points. See the related antiBrocard triangles. ——— Brocard (first) triangle A1B1C1 A1 = (a^2 : c^2 : b^2) is the projection of K = X(6) on the perpendicular bisector of BC. Also, A1 is the Psi image of the Avertex of the circumcevian triangle of X(2). See details in "Orthocorrespondence and Orthopivotal Cubics", §5 and also K018, K022. Also, A1 is the inverse of the midpoint of BC in the AMcCay circle. The GHirst inverses of A1, B1, C1 are the vertices a1, b1, c1 of the first antiBrocard triangle. ——— Brocard (second) triangle A2B2C2 A2 = (b^2 + c^2  a^2 : b^2 : c^2) = (2 SA : b^2 : c^2) is the projection of O = X(3) on the symmedian AK. Also, A2 is the Psi image of A. The Psi images of the sidelines of ABC are the McCay circles. These points A2, B2, C2 are the foci of the two sets of Artzt's parabolas whose directrices are the corresponding medians of ABC for the second set. A2 is also the inverse in (O) of the center Ωa of the AApollonius circle or, equivalently, the inverse of X(3) in the AApollonius circle. Recall that the eight points Ω1, Ω2, A1, B1, C1, A2, B2, C2 lie on the Brocard circle. ——— Brocard (third) triangle A3B3C3 A3 = (b^2c^2 : b^4 : c^4) is the isogonal conjugate of A1. A3 lies on the Acevian line of X(32), the 3rd power point, and also on the parallel at X(194) to the Acevian line of X(76). This latter line is the anticomplement of the line AX(76) and contains the Avertex of the antimedial triangle. A3 also lies on the line passing through X(2) and the center Ωa of the AApollonius circle. A3 is in fact the GHirst inverse of Ωa. More generally, the GHirst inverse of the Lemoine axis (which passes through Ωa, Ωb, Ωc) is an ellipse homothetic to the Steiner ellipses and passing through A3, B3, C3, X(2), X(194), Ω1, Ω2. Its center is X(7757), the midpoint of X(2)X(194), and its axes are parallel to those of the Steiner ellipses and to the asymptotes of the Kiepert hyperbola. This ellipse is mentioned by Randy Hutson, article X(7757) of ETC, in a totally different context. ——— Brocard (fourth) triangle A4B4C4 (sometimes called Dtriangle) A4 = (a^2 : b^2 + c^2  a^2 : b^2 + c^2  a^2) = (a^2 : 2 SA : 2 SA) is the isogonal conjugate of A2. A4 is the projection of H = X(4) on the median AG hence it lies on the orthocentroidal circle, also on the AApollonius circle and on the AMcCay circle. Also, A4 is the Psi image of the trace on BC of the trilinear polar of X(523), this latter line being the Psi transform of the orthocentroidal circle. Also, A4 is the orthoassociate of the trace of the orthic axis on BC. Recall that the orthoassociate (i.e. the inverse in the polar circle) of the orthic axis is the orthocentroidal circle. These points are the foci of the Brocard's parabolas whose directrices are the corresponding symmedians of ABC. *** A lot of other interesting properties can be found in :
*** In December 2015, Randy Hutson introduced two other Brocard triangles defined in ETC in the articles X(32) and X(384) respectively. Brocard (fifth) triangle A5B5C5 A5 = (a^2b^2 + a^2c^2 + b^2c^2 + b^4 + c^4 :  b^4 :  c^4). Brocard (sixth) triangle A6B6C6 A6 = (a^2b^2 + a^2c^2  b^2c^2 :  b^4 + c^4 + b^2c^2 : b^4  c^4 + b^2c^2). In September 2020, Dan Reznik and Peter Moses introduced another Brocard triangle namely Brocard (seventh) triangle A7B7C7 A7 = (a^4 + (b^2  c^2)^2 : 2 b^2 SB : 2 c^2 SC). See definitions, figures and properties at the bottom of this page. 



Centers in the Brocard triangles 

The following lists of centers were kindly provided by Peter Moses (20170314) Brocard 1 : {1,3923}, {2,2}, {3,182}, {4,1352}, {6,3734}, {10,3821}, {13,3642}, {14,3643}, {20,6776}, {22,184}, {23,110}, {25,9306}, {30,542}, {31,4112}, {32,4048}, {36,5150}, {69,2549}, {74,6795}, {75,3735}, {76,3094}, {98,3}, {99,6}, {100,5091}, {105,1083}, {110,1316}, {111,5108}, {114,5}, {115,141}, {125,11007}, {141,4045}, {147,4}, {148,69}, {183,574}, {187,5026}, {230,620}, {251,10328}, {262,7697}, {298,6772}, {299,6775}, {305,3981}, {316,11646}, {325,115}, {351,11183}, {376,11179}, {381,11178}, {383,5613}, {385,99}, {401,287}, {468,5972}, {511,2782}, {512,804}, {513,2787}, {514,2786}, {515,2792}, {516,2784}, {517,2783}, {518,2795}, {519,2796}, {520,2797}, {521,2798}, {522,2785}, {523,690}, {524,543}, {525,2799}, {530,531}, {531,530}, {538,5969}, {542,30}, {543,524}, {549,10168}, {560,4172}, {612,3980}, {614,4011}, {616,10654}, {617,10653}, {620,3589}, {649,4107}, {667,4164}, {669,5027}, {671,599}, {690,523}, {804,512}, {826,9479}, {850,3569}, {858,125}, {1078,5116}, {1080,5617}, {1194,4074}, {1281,1}, {1370,1899}, {1499,2793}, {1501,4159}, {1503,2794}, {1513,114}, {1691,5149}, {1799,10329}, {1916,76}, {1975,5028}, {1995,5651}, {2023,3934}, {2071,5622}, {2482,597}, {2770,9129}, {2782,511}, {2783,517}, {2784,516}, {2785,522}, {2786,514}, {2787,513}, {2788,3309}, {2789,3667}, {2790,6000}, {2791,6001}, {2792,515}, {2793,1499}, {2794,1503}, {2795,518}, {2796,519}, {2797,520}, {2798,521}, {2799,525}, {2857,8429}, {2996,10008}, {3006,3120}, {3146,5921}, {3263,3125}, {3266,3124}, {3268,1640}, {3309,2788}, {3314,7790}, {3407,10000}, {3413,3413}, {3414,3414}, {3543,11180}, {3667,2789}, {3705,3944}, {3747,4154}, {3757,846}, {3818,9996}, {3849,9830}, {3920,4418}, {3934,10007}, {3978,694}, {4027,384}, {4108,351}, {4226,5967}, {5026,7804}, {5121,11814}, {5152,1691}, {5159,6723}, {5182,11286}, {5189,3448}, {5205,1054}, {5913,126}, {5939,187}, {5969,538}, {5971,111}, {5976,39}, {5977,37}, {5978,13}, {5979,14}, {5980,15}, {5981,16}, {5982,17}, {5983,18}, {5984,20}, {5985,21}, {5986,22}, {5987,23}, {5988,10}, {5989,32}, {5990,100}, {5991,101}, {5992,8}, {5996,9148}, {5999,98}, {6000,2790}, {6001,2791}, {6031,353}, {6033,3818}, {6036,140}, {6054,381}, {6055,549}, {6108,619}, {6109,618}, {6114,624}, {6115,623}, {6194,7709}, {6233,6232}, {6323,6322}, {6636,5012}, {6660,3506}, {6721,3628}, {7061,3496}, {7391,11442}, {7426,5642}, {7464,11579}, {7492,11003}, {7610,7622}, {7622,7606}, {7697,11261}, {7710,7694}, {7711,7698}, {7774,11185}, {7778,7844}, {7779,148}, {7788,11648}, {7792,7820}, {7799,6034}, {7806,7835}, {7840,671}, {7868,7913}, {7931,7919}, {7983,5695}, {8178,8177}, {8289,3972}, {8290,83}, {8295,3406}, {8587,11149}, {8591,1992}, {8592,598}, {8593,11159}, {8594,12155}, {8595,12154}, {8596,11160}, {8597,11161}, {8724,5476}, {8782,194}, {8783,695}, {8857,3497}, {8858,3224}, {9067,11650}, {9147,5652}, {9148,11182}, {9149,5118}, {9185,1649}, {9191,8371}, {9469,3493}, {9478,6292}, {9479,826}, {9770,7615}, {9772,262}, {9828,3111}, {9829,10166}, {9830,3849}, {9855,8593}, {9861,6759}, {9865,1916}, {9866,11606}, {9870,9146}, {9877,11184}, {10011,6721}, {10163,10160}, {10352,7770}, {10418,11053}, {10717,9169}, {10989,9140}, {10992,1353}, {10997,4027}, {11161,5077}, {11177,376}, {11184,7617}, {11606,2896}, {11646,7761}, {11676,12177}, {11711,4672}, {12042,5092}, {12131,5907}, {12188,3098} where for example {3,182} means X(3) of Brocard 1 = X(182) of ABC. It can also be interpreted as X(182) of 1st antiBrocard = X(3) of ABC. Brocard 2 : {2,10166}, {3,182}, {6,574}, {15,15}, {16,16}, {110,9129}, {187,187}, {511,511}, {512,512}, {1340,11171}, {1379,6}, {1380,3}, {2028,8589}, {2029,39}, {2459,2459}, {2460,2460}, {3413,9830}, {3557,3094}, {5638,2502}, {5639,351}, {6566,6566}, {6567,6567} Brocard 3 : {2,11196} Brocard 4 : {2,6032}, {3,381}, {6,6}, {15,13}, {16,14}, {23,111}, {32,5309}, {39,7753}, {110,6792}, {111,2}, {182,5476}, {187,115}, {351,8371}, {352,9140}, {353,5640}, {511,542}, {512,690}, {543,3849}, {574,5475}, {843,9144}, {1296,4}, {1383,9465}, {1495,6791}, {1691,6034}, {1995,9745}, {2080,11632}, {2502,1648}, {2709,11005}, {2780,1499}, {2793,8704}, {2854,524}, {3098,3818}, {3231,8288}, {5008,5355}, {5104,11646}, {5107,5477}, {6088,523}, {6200,6564}, {6233,6785}, {6323,6787}, {6396,6565}, {8599,11215}, {8617,7703}, {8644,1637}, {8705,2854}, {9129,9169}, {9135,1640}, {9156,5466}, {9172,10162}, {9208,11182}, {9301,12188}, {9486,868}, {9870,6325}, {9871,74}, {9872,67}, {10354,427}, {10355,25}, {10765,1992}, {10787,9831}, {11186,3569}, {11258,3}, {11622,9175}, {11835,1327}, {11836,1328} Brocard 5 : {1,9941}, {2,7811}, {3,9821}, {4,9873}, {6,3094}, {13,9982}, {14,9981}, {54,9985}, {61,3104}, {62,3105}, {68,9923}, {74,9984}, {76,9983}, {83,2896}, {98,9862}, {99,8782}, {182,3098}, {384,76}, {485,9987}, {486,9986}, {671,9878}, {729,9998}, {1342,1670}, {1343,1671}, {1691,2076}, {2080,9301}, {3398,3}, {3407,10000}, {4027,99}, {5007,39}, {6655,7802}, {6656,7750}, {7760,194}, {7765,7756}, {7768,7893}, {7787,3096}, {7794,7826}, {7819,7767}, {7827,7833}, {7829,7830}, {7883,9939}, {7924,11057}, {10333,7768}, {10350,315}, {10788,9993}, {10789,11368}, {10790,11386}, {10791,9857}, {10794,10874}, {10795,10873}, {10796,9996}, {10797,10872}, {10798,10871}, {10799,11494}, {10800,9997}, {10803,10879}, {10804,10878}, {11303,9988}, {11304,9989}, {11364,3099}, {11380,10828}, {11490,10877}, {11636,9999}, {11839,11885}, {12110,4}, {12150,2}, {12176,98}, {12191,671}, {12192,74}, {12193,68}, {12194,1}, {12195,8}, {12196,84}, {12197,40}, {12198,80}, {12199,104}, {12200,7160}, {12201,265}, {12202,64}, {12203,3062}, {12204,14}, {12205,13}, {12206,83}, {12207,1297}, {12208,54}, {12209,10266}, {12210,486}, {12211,485}, {12212,6} Note that X(32) and all the points at infinity are the same in ABC and Brocard 5 since these are homothetic triangles at X(32). Brocard 6 : {2,7833}, {3,11257}, {4,9863}, {6,76}, {13,9988}, {14,9989}, {30,542}, {32,1975}, {83,2896}, {98,20}, {99,194}, {115,7750}, {141,7756}, {148,7893}, {182,3}, {511,2782}, {512,804}, {513,2787}, {514,2786}, {515,2792}, {516,2784}, {517,2783}, {518,2795}, {519,2796}, {520,2797}, {521,2798}, {522,2785}, {523,690}, {524,543}, {525,2799}, {530,531}, {531,530}, {538,5969}, {542,30}, {543,524}, {597,7810}, {671,9939}, {690,523}, {804,512}, {826,9479}, {1499,2793}, {1503,2794}, {1691,99}, {1692,5976}, {1916,9983}, {2456,98}, {2458,5989}, {2782,511}, {2783,517}, {2784,516}, {2785,522}, {2786,514}, {2787,513}, {2788,3309}, {2789,3667}, {2790,6000}, {2791,6001}, {2792,515}, {2793,1499}, {2794,1503}, {2795,518}, {2796,519}, {2797,520}, {2798,521}, {2799,525}, {3309,2788}, {3413,3413}, {3414,3414}, {3589,7830}, {3618,7904}, {3667,2789}, {3849,9830}, {4027,384}, {5026,39}, {5034,183}, {5038,1078}, {5149,3094}, {5182,2}, {5969,538}, {6000,2790}, {6001,2791}, {6033,9873}, {6034,7811}, {7809,9878}, {9426,887}, {9479,826}, {9830,3849}, {10131,10131}, {10352,7791}, {10353,7876}, {10791,4655}, {10800,5695}, {11606,9990}, {11646,7802}, {12151,671}, {12177,4}, {12213,13}, {12214,14}, {12215,1916}, {12216,11606} 



Similar triangles 

A1B1C1 is homothetic to the circumcevian triangles of X(3413), X(3414) with centers of homothety X(1341), X(1340) respectively. Recall that X(3413), X(3414) are the infinite points of the asymptotes of the Kiepert hyperbola and the axes of the Steiner ellipses. A1B1C1 is directly similar to the circumcevian triangle of any point P on the line at infinity. The center of similitude S is the inverse in the Brocard circle of the isogonal conjugate of P. 
A1B1C1 is indirectly similar to ABC and to the medial triangle. 

A2B2C2 is directly similar to the pedal triangle of X(23), the inverse of G in the circumcircle. 
A2B2C2 is indirectly similar to the pedal triangle of the centroid G. 

A3B3C3 is directly similar to the antipedal triangle of Zd, the isogonal conjugate of X(14870) hence A3B3C3 is directly similar to the pedal and circumcevian triangles of X(14870). 
A3B3C3 is indirectly similar to the pedal triangle of Zi, the inverse of X(14870) in the circumcircle. Zi lies on the lines X(3)X(14870) and X(32)X(76). A3B3C3 is indirectly similar to the antipedal triangle of the isogonal conjugate Zi* of Zi. This latter triangle is directly similar to the circumcevian triangle of Zi*. 

A4B4C4 is directly similar to the pedal triangle of X(187), the inverse of K in the circumcircle. 
A4B4C4 is indirectly similar to the antipedal triangle of the centroid G. 



Perspective triangles 

ABC and A1B1C1 are triply perspective at X(76), Ω1, Ω2. ABC and A2B2C2 are simply perspective at X(6). ABC and A3B3C3 are triply perspective at X(32), Ω1, Ω2. ABC and A4B4C4 are simply perspective at X(2). *** Two Brocard triangles are in general not perspective with two notable exceptions. 

A1B1C1 and A2B2C2 are simply perspective at G. The axis of perspective is the line passing through X(99), X(110). This is the trilinear polar of the isotomic conjugate of X(115), the center of the Kiepert hyperbola. 
A1B1C1 and A3B3C3 are triply perspective at X(384), Ω1 and Ω2. The corresponding axes of perspective are the lines L0, L1 and L2. L0 is the trilinear polar of X(385). L1 and L2 meet at X(694), the isogonal conjugate of X(385). 



Orthologic and parallelogic triangles 

ABC and A1B1C1 are orthologic at X(3) and X(98). ABC and A1B1C1 are parallelogic at X(6) and X(99). 



Loci related with perspective triangles 

The following table gives the locus of point P such that one of the Brocard triangles is perspective with one of the usual triangles related to P. See Table6. Some loci of corresponding perspectors are also given. 





Cubics through Brocard related points 

This section written with the collaboration of Chris van Tienhoven. The table below sums up all the catalogued cubics passing through the Brocard points Ω1, Ω2 and/or the vertices of some Brocard triangle. Those highlighted in orange contain the four foci of the Brocard ellipse namely Ω1, Ω2 and the common imaginary points of the Brocard axis and the Kiepert hyperbola. These two latter points also lie on K003, K049, K102, K115, K268, K369, K373, K587, K588, K643, K708, K731. The red cubics belong to a same pencil of stelloids. 



Notes : K053ABC are the three Apollonian strophoids. Each cubic contains only one vertex of the second Brocard triangle. K083ABC are the three equibrocardian focals. Each cubic contains only one vertex of the second Brocard triangle. K012* is nK(X32, X6, X2). It is the isogonal transform of K012. K023* is a circular cubic with focus X(98). It is the isogonal transform of K023. *** Circumcubics passing through the vertices of two Brocard triangles In general, there is one and only one such cubic (most of the time quite uninteresting) with the notable exception of the triangles A1B1C1 and A3B3C3. In this latter case, one can find a pencil of cubics generated by K017 and K020. This pencil is stable under isogonal conjugation i.e. the isogonal transform of the cubic passing through a point P is the cubic passing through P*. *** Circumcubics passing through the Brocard points and the vertices of one Brocard triangle In all four cases, these cubics form a pencil and must contain a nineth (fixed) point F. • with A1B1C1, F = X(76) and the pencil is generated by K012 and K512. • with A3B3C3, F = X(32) and the pencil is generated by K444 (the isogonal transform of K512) and the isogonal transform of K012 namely nK(X32, X6, X2). Note that the isogonal transform of a cubic of the former pencil is a cubic of the latter pencil and vice versa. • with A2B2C2 and A4B4C4, the corresponding points F do not appear in the present version of ETC but the note above remains true. *** Other remarkable circumcubics The cubics K019, K021, K023 are all circular cubics of a same pencil with the same focus X(98) and passing through the Brocard points. The real asymptote passes through X(99) and the orthic line passes through X(3). It follows that the polar conic of X(3) in each cubic is a rectangular hyperbola whose center is a point on the circle with diameter X(4)X(147). Once again, this pencil is stable under isogonal conjugation and contains the isogonal transform of K023. See figure below and also CL056. 



The 5th and 6th Brocard triangles 

Let a1b1c1 and a2b2c2 be the circumcevian triangles of the Brocard points Ω1 and Ω2 respectively. The lines a1a2, b1b2, c1c2 bound a triangle A5B5C5 called the 5th Brocard triangle. A5B5C5 is homothetic to • ABC at X(32), ratio : 4 cos^{2}ω  1, • the medial triangle at X(3096), • the antimedial triangle at X(2896). It is perspective to • the 1st, 3rd Brocard triangles at X(2896), X(32) respectively, • the cevian triangle of any P on pK(X3051, X141) with perspector on pK(X32 x X3096, X2896), • the anticevian triangle of any P on pK(X32 x X3096, X2) with perspector on pK(X32 x X3096, X2896). 

The (red) second Brocard circle is the circle with center X(3) passing through the Brocard points Ω1, Ω2. The lines AΩ1, AΩ2 meet this circle again at a1, a2 respectively. The points b1, b2 and c1, c2 are defined similarly. The lines a1a2, b1b2, c1c2 bound a triangle A6B6C6 called the 6th Brocard triangle. A6B6C6 is homothetic to • the 1st Brocard triangle at X(384), ratio : 4 cos^{2}ω, • the 1st antiBrocard triangle at X(3), ratio : 4 cos^{2}ω  3. It is perspective to the 3rd Brocard triangle at X(384) and indirectly similar to ABC. The circumcircle of A6B6C6 is the circle with diameter X(20)X(194). 

These two triangles A5B5C5 and A6B6C6 are perspective at P = X(9983) with SEARCH = 11.9474600405803. P lies on the lines X(20)X(2782), X(32)X(76), X(39)X(3096), X(69)X(194), etc. Its abscissa in X(32), X(76) is : 4 cos^{2}ω  1 and its abscissa in X(194), X(69) is : cos2ω.
A5B5C5 and A6B6C6 are obviously indirectly similar since A5B5C5, ABC are homothetic and A6B6C6, A1B1C1 are also homothetic, this latter triangle being indirectly similar to ABC. 



The 7th Brocard triangle 

A7 is the second intersection of the line OA with the Brocard circle. It is analogous to A2 where KA is replaced by OA. A7 is also : • the inverse in the circumcircle of the trace of OA on the Lemoine axis (trilinear polar of the symmedian point K). Recall that A2 is the inverse in the circumcircle of the trace Ωa on BC. See generalization 2 below. • the Psi image of the Avertex of the circumcevian triangle of X(25). With X(2) instead of X(25), we find A1. See generalization 1 below. A7B7C7 is perspective • at O to ABC and obviously to any triangle whose vertices lie on the cevian lines of O respectively. • at X(184) to A2B2C2. • to the cevian triangle of P for every P on a pK passing through X(i) for i = 3, 6, 230, 5254, 6530, 8770. Its pole is {X3,X114}/\{X115,X2353} and its isopivot is O. • to the anticevian triangle of P for every P on pK(X3 x X3767, X230) passing through X(i) for i = 3, 69, 230, 248, 2450, 3767. A7B7C7 is directly similar to the pedal triangle of X(186) and indirectly similar to the orthic and tangential triangles. A7B7C7 is parallelogic to the cevian triangle of P for every P on a circumcubic passing through X(i) for i = 4, 670, 850. A7B7C7 is orthologic to the cevian triangle of P for every P on a circumcubic passing through X(i) for i = 4, 68, 290, 401. A7B7C7 is orthologic to the pedal triangle of P for every P on the line passing through X(i) for i = 4, 69, 76, 264, 286, 311, 314, 315, 316, 317, 340, 511, etc. One center of orthology lies on the hyperbola passing through A7, B7, C7, X(3), X(184) and the other lies on the line passing through X(i) for i = 114, 325, 511, 1513, 2679, 2683, 2684, etc.




Generalizations 

Generalization 1 The Psi image QaQbQc of the circumcevian triangle PaPbPc of any point P = u:v:w is inscribed in the Brocard circle. QaQbQc is perspective to A2B2C2 at Q = 2 SA u + a^2 (v + w) : : . With P = X(2), QaQbQc is A1B1C1 and Q = X(2). With P = X(25), QaQbQc is A7B7C7 and Q = X(184). When P lies on the Lemoine axis, QaQbQc and A2B2C2 are triply perspective with perspectors also on the Lemoine axis. P and Q coincide when P = X(2) already mentioned and when P is X(5638) or X(5639). These two points lie on the Lemoine axis, the Parry circle, the ThomsonJerabek hyperbola. QaQbQc and ABC are generally not perspective unless P lies on a circumquartic passing through X(2), X(3), X(25), X(111) in which case the perspector is X(76), X(6090), X(3), X(6) respectively. 

Generalization 2 Let P be a point not lying on the Lemoine axis and denote by Pa, Pb, Pc the traces of the cevian lines of P on the Lemoine axis. The inverses Qa, Qb, Qc in the circumcircle of Pa, Pb, Pc obviously lie on the Brocard circle. ABC and QaQbQc are perspective when P lies on a circular circumquintic passing through X(3), X(25), X(32), X(187), X(1379), X(1380), the cevians of X(187), the infinite points of pK(X14575 = X3 x X32, X187). With P = X(3), X(32), we find A7B7C7, A1B1C1 respectively. 
