KöMaL 2018 B4965

For a vector $\mathbf x\ne \mathbf 0$, let $\mathbf e_{\mathbf x}=\frac{\mathbf x} {|\mathbf x|}$. A given plane $\mathcal{S}$ is parallel, but not identical to the plane of a given (non-degenerate) triangle $ABC$. Show that there exists a unique $P\in \mathcal{S}$, such that the vector $\mathbf e_{\overrightarrow {PA}}+\mathbf e_{\overrightarrow {PB}}+\mathbf e_{\overrightarrow {PC}}$ is perpendicular to $\mathcal{S}$.