We are given a triangle $ABC$ with $G$ as the centroid, $O$ as the circumcenter, and $H$ as the orthocenter. The lengths $OG = 9$ and $OH = 15$ are provided, and we need to find the length of $GH$.

In a triangle, the points $O$ (circumcenter), $G$ (centroid), and $H$ (orthocenter) are collinear and lie on a line called the Euler line. The centroid $G$ divides the Euler line in the ratio 2:1, with the segment from the orthocenter $H$ to the centroid $G$ being twice the length of the segment from the centroid $G$ to the circumcenter $O$.

Let the length of $GH$ be $x$. Since $OG = 9$, and the centroid divides the Euler line in the ratio 2:1, we can set up the following relationship:

$GH = 2 \cdot OG = 2 \cdot 9 = 18$

Now, using the total length of the Euler line, we have:

$OH = OG + GH$

Substituting the known values:

$15 = 9 + 18$

However, this is a contradiction since $15 \neq 27$, so we need to revisit our assumption that $GH = 18$. Let me review this more carefully.

The solution for $GH$