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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 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^2c^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^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).

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 (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}

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

TR1dir TR1indir

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.

TR2dir TR2indir

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.

TR3dir TR3indir

A3B3C3 is directly similar to the antipedal triangle of Zd, a point with complicated coordinates, SEARCH = 7.21144473704106.

A3B3C3 is indirectly similar to the pedal triangle of Zi, a point on the line X(32)X(76) with complicated coordinates, SEARCH = 2.90316647259929.

TR4dir TR4indir

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.

TR1TR2persp TR1TR3persp

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.

 

A1B1C1

A2B2C2

A3B3C3

A4B4C4

cevP

K322 = pK(X1916, X694)

K531 = pK(X3455, X67)

K532 = pK(X8789, X694)

K533 = pK(X8791, X8791)

acvP

K128 = pK(X6, X385)

K534 = pK(X3, X524)

K128 = pK(X6, X385)

K535 = pK(X25, X468)

pedP

a conic through X(3), X(3098)

a cubic

a cubic

a cubic through O

apdP

K422 = pK(X6, X5999)

K536, a circular cubic

K422 = pK(X6, X5999)

K537, a circular cubic

ccvP

a quartic through X(1), X(76), X(694)

a quartic through X(6), X(2930)

a quartic through X(6), X(32)

a quartic through X(2), X(67)

cacP

a sextic through X(1), X(6)

a sextic through X(6)

a sextic through X(6)

a sextic through X(1), X(67)

refP

a cubic through X(3), X(511), X(2076)

a cubic through X(184)

a cubic

a cubic through X(935)

loci of perspectors

cevP acvP

K020 = pK(X6, X384)

K538 = pK(X3, X141)

K020 = pK(X6, X384)

K539 = pK(X25, X427)

 

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.

cubic

type / name

Ω1, Ω2

A1B1C1

A2B2C2

A3B3C3

A4B4C4

X(i) on the cubic for i =

K012

isotomic nK / Tucker-Brocard

 

 

 

6, 76, 880, 882

K017

isogonal nK0 / Brocard 1

 

 

 

2, 6, 99, 512

K018

isogonal nK0 / Brocard 2

 

 

 

2, 6, 13, 14, 15, 16, 111, 368, 524

K019

isogonal focal nK0 / Brocard 3

 

 

 

 

98, 511

K020

isogonal pK / Brocard 4

 

 

 

1, 3, 4, 32, 39, 76, 83, 194, 384, 695, 2896, 3224, 3491 up to 3503

K021

isogonal pK / Brocard 5

 

 

 

 

1, 99, 512, 2142, 2143

K022

nK / Brocard 6

 

 

 

 

3, 4, 110, 525

K023

orthopivotal cubic / Brocard 7

 

 

 

 

4, 13, 14, 30, 1316

K083

see note below

 

 

 

 

368

K153

- / Brocard 8

 

 

 

 

6

K166

isogonal focal nK / Brocard 9

 

 

 

 

3, 4, 1316, 2698, 2782

K248

isogonal nK / Brocard-Steiner focal

 

 

 

 

187, 538, 671, 729

K326

isogonal pK / Brocard 10

 

 

 

 

1, 39, 83

K359

isogonal focal nK / Brocard strophoid

 

 

 

 

1, 1083

K444

psK / K(X32)

 

 

 

4, 32, 194

K509

Kiepert strophoid

 

 

 

 

2, 13, 14, 125, 543, 1316, 5465, 9169

K512

psK / Brocard-van Tienhoven

 

 

 

3, 76, 3224

K516

K60+ / Brocard-McCay

 

 

 

 

4, 3095

K538

pK / -

 

 

 

 

6, 69, 141, 1176, 2916

K539

pK / -

 

 

 

 

2, 25, 251, 427

K012*

nK / -

 

 

 

2, 32, 881

K023*

isogonal transform of Brocard 7

 

 

 

 

3, 15, 16, 74

K545

K0+ / Brocard 11

 

 

 

76

K546

- / Brocard 12

 

 

 

32

K547

- / Brocard 13

 

 

 

39, 76

K548

- / Brocard 14

 

 

 

76, 2896

K549

K+ / Brocard 15

 

 

 

 

K550

- / Brocard 16

 

 

 

76

K551

- / Brocard 17

 

 

 

76

K552

- / Brocard 18

 

 

 

76, 99

K553

acnodal cubic / Brocard 19

 

 

 

2, 76

K629

psK / -

 

 

 

 

54, 112, 115, 826

K630

psK / -

 

 

 

 

5, 249, 525, 827

K688

isogonal nK / -

 

 

 

 

2, 6, 5969, 5970

K793

circular nodal cubic

 

 

 

 

2, 6, 111, 1340, 1341, 5622, 5969

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.

K019K021K023

 

The 5th and 6th Brocard triangles

TR5

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).

TR6

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).

TR56

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.