In triangle $ABC$, we have the centroid $G$, circumcenter $O$, and orthocenter $H$. Given that $OG = 9$ and $OH = 15$, we need to find the length of $GH$.

The points $O$, $G$, and $H$ lie on the Euler line of the triangle, and the centroid $G$ divides the Euler line in the ratio $2:1$ such that:

$OG : GH = 2 : 1$

Given: $OG = 9$

Let $GH = x$. Then by the ratio:

$\frac{OG}{GH} = \frac{2}{1} \implies \frac{9}{x} = \frac{2}{1}$

Cross-multiplying, we find:

$9 = 2x \implies x = \frac{9}{2} = 4.5$

Thus, $GH = 4.5$.

Final Answer:

$GH = 4.5$

Would you like a detailed explanation, or do you have any questions? Here are 5 related questions that you might find interesting:

1. How is the centroid related to the medians of a triangle?
2. What are the properties of the circumcenter in different types of triangles?
3. Can the Euler line be absent in certain triangles? If so, which ones?
4. How does the orthocenter's position change based on the type of triangle?
5. What is the relationship between the circumradius and the distances between the triangle's key points?

Tip: Remember that the Euler line contains several key points of a triangle, such as the orthocenter, centroid, and circumcenter, and understanding its properties can help solve various geometric problems.