 Solution

Taking the dot product with $\mathbf{n}$ gives $\mathbf{n}\cdot \mathbf{r}=\lambda +\mu +\nu$. The necessary and sufficient condition is thereforeThese 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.