A relation between the Brocard and Miquel Angles

Kiss, Sándor Nagydobai

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DOI: https://doi.org/10.37193/CMI.2020.02.06

Issue no: Vol 29/2020 no. 2 Tags: Miquel angle, Brocard angle, barycentric coordinates

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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 $ABC$ .