The Brocard Angle and a Geometrical Gem from Dmitriev and Dynkin

In a celebrated paper on the eigenvalues of special matrices, N. Dmitriev and E. Dynkin formulated and proved a nice geometrical lemma. This lesser-known gem provides a simple proof of the maximal value of the Brocard angle and yields also the solution to some challenging problems from the past few decades. The Brocard points and Brocard angle of a triangle have attracted great attention since their discovery in the beginning of the 19th century. Many properties and their generalizations to polygons of these geometrical objects appear from time to time in various forms such as mathematical olympiad problems or challenging problems in expository mathematical journals. One frequently occuring question is the maximal value of the Brocard angle. In this note, we make an attempt to collect some of this problem’s appearances and to link them to a hidden geometrical lemma from a 1945 paper on the location of eigenvalues of special matrices in the complex plane. The Brocard Points and Angles The history of the Brocard points began in 1816 when August Leopold Crelle (1780–1855) published a small book on plane triangles. He was dealing with a certain point P inside a triangle ABC such that the angles ^PAB, ^PBC, ^PCA are equal, see Fig. 1. Among other things, he showed that (for a given oder of vertices) there uniquely exists such a point P and the angle χ = ^PAB = ^PBC = ^PCA satisfies (1) χ = cot^CAB + cot^ABC + cot^BCA. By reversing the order of vertices we obtain another point P ′ with the same angle χ, see Fig. 1. These remarkable points of a triangle were studied some years after Crelle by Carl Friedrich Andreas Jacobi (1801–1875) who should not to be confused with the perhaps much famous Carl Gustav Jakob Jacobi (as some might do, by references not mentioning the first names of Jacobi). However, the points were named after the French geometer and military officer Henri Brocard (1845–1922) who rediscovered them in 1875 and drew again considerable attention to explore their many fascinating properties (see [7, 8] for a detailed exposition on this topic). Such a property, which is usually derived from the formula (1), is that the Brocard angle χ has a maximum value,