Let XYZ be an inscribed triangle in the reference triangle ABC (X\in BC, Y\in CA, Z\in AB) given by its absolute barycentric coordinates: X=(0,1-\alpha,\alpha), Y=(\beta,0,1-\beta), Z=(1-\gamma,\gamma,0), where \alpha,\beta,\gamma are arbitrary real numbers. In this paper we deduce the following simple and beautiful formula between the Brocard and Miquel angles: S_\omega=S_\theta+a^2\alpha+b^2\beta+c^2\gamma, where \omega is the Brocard and \theta the Miquel angle of ABC.





Additional Information


Kiss, Sándor Nagydobai