Brocard points

The most elegant construction of the Brocard points goes as follows.

In geometry, Brocard points are special points within a triangle.

The Brocard points are isogonal conjugates of each other.

It contains the Brocard points.

The 1st equal areas cubic passes through the incenter, Steiner point, other triangle centers, the 1st and 2nd Brocard points, and the excenters.

Several binary operations, such as midpoint and trilinear product, when applied to the two Brocard points, as well as other bicentric pairs, produce triangle centers.

This strict definition exclude the excentres, and also excludes pairs of bicentric points such as the Brocard points (which are interchanged by a mirror-image reflection).

Brocard's most well-known contributions to mathematics are the Brocard points, the Brocard circle, and the Brocard triangle.

Henri Brocard introduces the Brocard points, Brocard triangle and Brocard circle.

His best-known achievement is the invention and discovery of the properties of the Brocard points, the Brocard circle, and the Brocard triangle, all bearing his name.

It rules out various well-known points such as the Brocard points, named after Henri Brocard (1845-1922), which are not invariant under reflection and so fail to qualify as triangle centers.

The two Brocard points are closely related to one another; In fact, the difference between the first and the second depends on the order in which the angles of triangle ABC are taken.

The first and second Brocard points are one of many bicentric pairs of points, pairs of points defined from a triangle with the property that the pair (but not each individual point) is preserved under similarities of the triangle.

The Brocard points are an example of a bicentric pair of points, but they are not triangle centers because neither Brocard point is invariant under similarity transformations: reflecting a scalene triangle, a special case of a similarity, turns one Brocard point into the other.