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Special points in barycentric coordinates

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Here, $(u, v, w)$ often refers to $\frac{1}{u + v + w}(u, v, w)$, for simplicity.

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Let $a = BC, b = AC, c = AB$.

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The symmedian point's coordinates are $(a^2, b^2, c^2)$. The coordinates of the orthocenter and the circumcenter are $(\tan A, \tan B, \tan C)$ and $(\sin 2A, \sin 2B, \sin 2C)$, and they can be proved by the area definition.

These are the coordinates of common special points.