[ Trilinear coordinates]

Find the trilinear coordinates of the Brocard points.

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Let $(x: y: z)$ be the trilinear coordinates of the first Brocard point $P$. Then $x: y: z = CP: AP: BP$. Moreover, $AP / \sin \varphi = AB / \sin \alpha = 2Rc / a$ (here $\varphi$ is the Brocard angle). Similarly, $BP = 2R \sin \varphi a / b$ and $CP = 2R \sin \varphi b / c$. Thus, the first Brocard point has trilinear coordinates $\left(\frac{b}{c}: \frac{c}{a}: \frac{a}{b}\right)$. The second Brocard point has trilinear coordinates $\left(\frac{c}{b}: \frac{a}{c}: \frac{b}{a}\right)$.