how to find the barycentric coordinates of the orthocenter $A = (0,0),B = (4,0),C = (1,2)$
How can I find the barycentric coordinates of the orthocenter of $\triangle ABC$?

This Wikipedia page describes the orthocentre in barycentric coordinates of the triangle $ABC$ as
$$
\left((a^2+b^2-c^2)(a^2-b^2+c^2):(a^2+b^2-c^2)(-a^2+b^2+c^2):(a^2-b^2+c^2)(-a^2+b^2+c^2)\right),
$$ where $a=|BC|,b=|CA|$ and $c=|AB|$ are the side lengths of the triangle.
