Problem
GEO-B3-M06-P019 Brocard Angle Formula
Let \(\varphi\) be the Brocard angle of triangle \(ABC\). Prove \(\operatorname{ctg}\varphi=\operatorname{ctg}A+\operatorname{ctg}B+\operatorname{ctg}C\).
C. Hint 1. Split area \(ABC\) into three triangles with vertex at the Brocard point.
D. Hint 2. Express distances through one common quantity and \(\varphi\).
Let \(P\) be the first Brocard point, \(\angle ABP=\angle BCP=\angle CAP=\varphi\). Split area \(S\) into \(S_{ABP}+S_{BCP}+S_{CAP}\). Each of these triangles has one angle \(\varphi\), and the neighboring angles are tied to \(A,B,C\).
Using the sine rule in triangles \(ABP\), \(BCP\), \(CAP\), express the ratios \(AP:BP:CP\), then substitute into the area sum. After canceling a common factor, one obtains \[ \operatorname{ctg}\varphi=\operatorname{ctg}A+\operatorname{ctg}B+\operatorname{ctg}C. \] The idea is that the three small areas share \(\sin\varphi\), while the cosine terms collect into the cotangent sum.
A good advanced example of area-and-sine computation.