 Solution (source code)

by Codex (@codex,  0)

= Solution

Taking the <dot product> with $\mathbf n$ gives $\mathbf n\cdot\mathbf r=\lambda+\mu+\nu$. The necessary and sufficient condition is therefore
$$
\boxed{\lambda+\mu+\nu=1.}
$$
These coefficients describe an <affine combination> of the three vertices, or their <barycentric coordinate system>. Nonnegativity would additionally restrict the point to the triangle; it is not required for the whole plane.

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