Home page | Catalogue | Classes | Tables | Glossary | Notations | Links | Bibliography | Thanks | Downloads | Related Curves

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 anti-Brocard 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 A-vertex 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 A-McCay circle.

The G-Hirst inverses of A1, B1, C1 are the vertices a1, b1, c1 of the first anti-Brocard 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 A-Apollonius circle or, equivalently, the inverse of X(3) in the A-Apollonius 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^2 c^2 : b^4 : c^4) is the isogonal conjugate of A1. A3 lies on the A-cevian line of X(32), the 3rd power point, and also on the parallel at X(194) to the A-cevian line of X(76). This latter line is the anticomplement of the line AX(76) and contains the A-vertex of the antimedial triangle.

A3 also lies on the line passing through X(2) and the center Ωa of the A-Apollonius circle. A3 is in fact the G-Hirst inverse of Ωa.

More generally, the G-Hirst 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 D-triangle)

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 A-Apollonius circle and on the A-McCay 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 :

• Lalesco T. : La géométrie du triangle, Gabay, Paris, 1987.
• Honsberger, R. : Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Washington, DC Math. Assoc. Amer., 1995.
• Johnson, R. A. : Advanced Euclidean Geometry, Dover Publications Inc, Mineola, New York, 1960.

***

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^2 b^2 + a^2 c^2 + b^2 c^2 + b^4 + c^4 : - b^4 : - c^4).

Brocard (sixth) triangle A6B6C6

A6 = (a^2 b^2 + a^2 c^2 - b^2 c^2 : - b^4 + c^4 + b^2c^2 : b^4 - c^4 + b^2c^2).

***

In September 2020, Dan Reznik and Peter Moses introduced three new Brocard triangles namely

Brocard (seventh, eighth, nineth) triangles A7B7C7, A8B8C8 (degenerate), A9B9C9

A7 = (a^4 + (b^2 - c^2)^2 : 2 b^2 SB : 2 c^2 SC), on the Brocard circle.

A8 = (-a^4 : b^2 SB : c^2 SC), inverse of A7 in the circumcircle.

A9 = (-SB SC : a^2 SC : a^2 SB), isogonal conjugate of A8.

In June 2021, Clark Kimberling and Peter Moses introduced another Brocard triangle – see preamble just before X(43803) – namely

Brocard (tenth) triangle A10B10C10

A10 = reflection of A7 in X(182), on the Brocard circle.

A10 = a^8 - 2 a^4 b^4 + 2 a^2 b^6 - b^8 - 2 a^2 b^4 c^2 + 4 b^6 c^2 - 2 a^4 c^4 - 2 a^2 b^2 c^4 - 6 b^4 c^4 + 2 a^2 c^6 + 4 b^2 c^6 - c^8 : b^2 ( - a^6 + a^4 b^2 - a^2 b^4 + b^6 - 3 a^4 c^2 - 3 b^4 c^2 + a^2 c^4 + 3 b^2 c^4 - c^6) : c^2 ( - a^6 - 3 a^4 b^2 + a^2 b^4 - b^6 + a^4 c^2 + 3 b^4 c^2 - a^2 c^4 - 3 b^2 c^4 + c^6)

***

Brocard (eleventh) triangle A11B11C11

A11 = Psi image of the antipode of A on the circumcircle, on the Brocard circle.

A11 = a^6 + a^4 b^2 - 3 a^2 b^4 + b^6 + a^4 c^2 + 6 a^2 b^2 c^2 - b^4 c^2 - 3 a^2 c^4 - b^2 c^4 + c^6 : b^2 (a^4 - 4 a^2 b^2 + 3 b^4 + 6 a^2 c^2 - 4 b^2 c^2 + c^4) : c^2 (a^4 + 6 a^2 b^2 + b^4 - 4 a^2 c^2 - 4 b^2 c^2 + 3 c^4)

For all these triangles, see other definitions, further properties and figures below in this page.

Centers in the Brocard triangles

The following lists of centers were kindly provided by Peter Moses (2017-03-14)

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 anti-Brocard = 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}, {4,31960}

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 to 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 in the cells highlighted in orange contain the four foci of the Brocard ellipse namely Ω1, Ω2 and the common imaginary points X(39641), X(39642) of the Brocard axis and the Kiepert hyperbola.

These two latter points also lie on the cubics (not passing through the Brocard points) K003, K049, K102, K115, K268, K369, K373, K587, K588, K643, K708, K731, K763, K801, K828, K833, K1096, K1098, K1113, K1139, K1200, K1283, and on the curves Q019, Q039, Q065, Q073, Q088, Q094, Q138, Q157, Q158, Q167, Q168, Q171.

The red cubics above belong to a same pencil of stelloids. See Table 51.

Notes :

K053-A-B-C are the three Apollonian strophoids. Each cubic contains only one vertex of the second Brocard triangle.

K083-A-B-C 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.

***

Circum-cubics 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*.

***

Circum-cubics 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 circum-cubics

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 cos²ω - 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).

A5B5C5 is orthologic to

• ABC at X(4) and X(9873).

• A1B1C1 at X(3) and X(9862).

A5B5C5 is parallelogic to

• A1B1C1 at X(6) and X(8782).

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 cos²ω,

• the 1st anti-Brocard triangle at X(3), ratio : 4 cos²ω - 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).

A6B6C6 is orthologic to

• A1B1C1 at X(1352) and X(9863).

A6B6C6 is parallelogic to

• ABC at X(99) and 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 cos²ω - 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.

A5B5C5 and A6B6C6 are orthologic at X(20) and X(9862), hence they are bilogic triangles.

A5B5C5 and A6B6C6 are parallelogic at X(194) and X(8782).

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 A8 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 A-vertex 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 circum-cubic passing through X(i) for i = 4, 670, 850.

A7B7C7 is orthologic to A1B1C1 (at X182 and X6776) and to A6B6C6 (at X11257 and X6776).

A7B7C7 is orthologic to the cevian triangle of P for every P on a circum-cubic 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.

The 8th, 9th, 10th Brocard triangles

These triangles are related to A7B7C7 as follows :

• A8, B8, C8 are the inverses of A7, B7, C7 in the circumcircle. These points are collinear on the Lemoine axis.

A8B8C8 is perspective (at Q) to the cevian (resp. anticevian) triangle of P for every P on pK(X32, X25) = K171 (resp. pK(X14575, X6) = K1313). In both cases, Q lies on pK(X14575, X3053).

***

• A9, B9, C9 are the isogonal conjugates of A8, B8, C8. These points lie on the Steiner ellipse.

A9B9C9 is perspective at X4, X458, X39646 to ABC, A2B2C2, A7B7C7 respectively.

A9B9C9 is orthologic to ABC (at X4 and X6776) and to A5B5C5 (at X4 and X39882).

***

• A10, B10, C10 are the antipodes of A7, B7, C7 in the Brocard circle.

A10B10C10 is perspective at X1352 to A1B1C1 and homothetic at X182 to A7B7C7.

A10B10C10 is orthologic to A1B1C1 (at X182 and X1352), to A6B6C6 (at X111257 and X1352), to A7B7C7 (at X39910 and X48716).

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) : : . Recall that A2B2C2 is the Psi image of ABC.

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 Thomson-Jerabek hyperbola and K889, K890, K891, K1067.

QaQbQc and ABC are generally not perspective unless P lies on a circum-quartic passing through X(2), X(3), X(25), X(111), X(5638) X(5639) in which case the perspector is X(76), X(6090), X(3), X(6), X(1341), X(1340) respectively. With P = X(3), we obtain the Brocard (eleventh) triangle, see below. The locus of the perspector is a very complicated circular circum-curve of degree 10 where A, B, C are quadruple points. It contains the points listed above, also A2, B2, C2 and the barycentric products of X(6) and the circular points at infinity which lie on the Lemoine axis, together with X(5638) X(5639).

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 circum-quintic 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). The locus of the perspector is the same curve of degree 10 as above.

With P = X(3), X(32), we find A7B7C7, A1B1C1 respectively. With P = X(25), we obtain the Brocard (eleventh) triangle, see below.

The 11th Brocard triangle

A11B11C11 is the Psi image of the circumcevian triangle A'B'C' of O. It is inscribed in the Brocard circle hence its circumcenter is X(182).

Its Lemoine point is X(61631). Its isodynamic points are X(61632), X(61633) on the Parry circle, the Psi images of X(14539), X(14538) respectively.

A11 is also the circumcircle inverse of the trace of the A-cevian line of X(25) on the Lemoine axis.

This triangle A11B11C11 is

• perspective to ABC, A2B2C2, A10B10C10 at X(6090), X(5651), X(9303) respectively.

• perspective to the circumcevian triangle of X(524) at the homothetic Z of X(1899) under h(X25, 1/3). Z lies on the lines {2, 51}, {4, 74}, {5, 3066}, {6, 468}, {25, 1503}, {64, 1906}, etc, SEARCH = 1.15364665239210.

• perspective to the anticevian triangle of X(524) at Z' on the lines {2, 154}, {5, 1511}, {53, 450}, {141, 468}, etc, SEARCH = 1.77934309780989.

• perspective to the circumcevian triangle of X(2930) at a rather complicated point, SEARCH = 2.28039183135249.

• directly similar to the circumcevian triangle of X(7464). The simili-center S lies on the lines {74, 182}, {112, 574}, etc, SEARCH = 4.64707280918433.

• orthologic and inversely similar to the pedal and circumcevian triangles of X(376). In the latter case, the similitude center is X(7709).

• parallelogic to the circumcevian triangle of X(376).

These points Z, Z', S are now X(61506), X(61507),X(61508) in ETC (2024-02-07).