Prove that among all possible triangles whose vertices are $3,5$ and $7$ apart from a given point $P$, the ones with the largest perimeter have $P$ as incentre.
Supose ABC is that triangle. Let $\Sigma$ the elipse of the points $Q$ such that $AQ+QB=AC+BC$, and $\Gamma$ the circle of centre $P$ and radius $7$. $C$ is the unique intersection point of $\Sigma$ and $\Gamma$ ( because the choice of $ABC$). In particular , there is a comun tangent $\iota$ to $\Sigma$ and $\Gamma$ passing by $C$. Let $X$ and $Y$ points in $\iota$ such that $C$ is betwen them. For property of the elipse we have $<XCA=<YCB$. Since $\iota$ is tangent to $\Gamma$ then $<ACP=<BCP$.