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.

Brocard (first) triangle A1B1C1

A1 = (a^2 : c^2 : b^2) is the projection of K = X(6) on the perpendicular bisector of BC.

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.

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.

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.

A lot of 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.

Similar triangles

A1B1C1 is directly similar to ?

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, a point with complicated coordinates.

A3B3C3 is indirectly similar to the pedal triangle of Zi, a point with complicated coordinates.

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

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).

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.

Notes :

K083-A-B-C are the three equibrocardian focals. Each cubic contains only one vertex of the second Brocard triangle.

K012* is the isogonal transform of K012 namely nK(X32, X6, X2).

***

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.