We show that if the point $P$ lies outside the segment $QR$, then the conditions of the problem are not satisfied.
If $P$ lies on the line $QR$ but outside the segment $QR$ (Fig. 3), then we can

Fig. 3

Fig. 4

take the point $X$ on the line $QR$ on the other side of the segment $QR$. Then the points $Q$ and $R$ are closer to the point $X$ than the point $P$.
If $P$ lies outside the line $QR$ (Fig. 4), then the perpendicular bisectors of the segments $PQ$ and $PR$ intersect. Choose the points $X$ in the region, which lies towards $Q$ from the perpendicular bisector of $PQ$ and towards $R$ from the perpendicular bisector of $PR$ (the dark region on the figure). Then the points $Q$ and $R$ are closer to the point $X$ than the point $P$.