1. **State the problem:** We need to find the value of $x$ such that point $N$ is the incenter of triangle $QRP$. The incenter is the point where the angle bisectors intersect and is equidistant from all sides of the triangle. 2. **Given information:** - $GN = 14x$ - $NP = 25$ - $HP = 24$ 3. **Key property of the incenter:** The incenter is equidistant from all sides of the triangle. Since $N$ lies on the angle bisectors, the distances from $N$ to each side are equal. 4. **Analyze the segments:** - $GN$ and $NP$ are parts of a segment along the angle bisector from $G$ to $P$. - Since $N$ is the incenter, the distances from $N$ to sides $RP$ and $QP$ must be equal. 5. **Set up the equation:** Since $HP = 24$ is the distance from $H$ to $P$, and $NP = 25$ is given, and $GN = 14x$, the total length $GP = GN + NP = 14x + 25$. 6. **Use the property of the incenter:** The distances from $N$ to the sides are equal, so $GN = HP$. 7. **Solve for $x$:** $$14x = 24$$ $$x = \frac{24}{14} = \frac{12}{7}$$ **Final answer:** $$x = \frac{12}{7}$$