In this geometric problem, we are working with a triangle $ABC$, where certain midpoints are given and a specific length is known. Let’s break down the information to solve for the length of $GH$.

Step-by-Step Analysis:

1. Key Points Defined:

   • $M$ is the midpoint of $BC$, so $BM = MC$.
   • $N$ is the midpoint of $MC$, so $MN = NC$.
   • $E$ is the midpoint of $AC$, so $AE = EC$.

2. BE as a Median:
   The line $BE$ is the median from vertex $B$ to side $AC$. It's given that $BE = 42 \, \text{cm}$.

3. Relationship Between GH and BE: From the figure, $G$ and $H$ are points on the line $AM$, which connects the vertex $A$ to the midpoint $M$ of side $BC$, and $N$ is the midpoint of $MC$.

   In such geometric configurations, $GH$ often represents a portion of the medians, particularly when working with centroid-related geometry. In triangle centroid properties:

   • The centroid divides each median into a 2:1 ratio.
   • Here, if $G$ is the centroid, then $H$ would typically lie along the median $AM$, and $GH$ would be one-third the length of the median from vertex $B$ (i.e., $BE$).

4. Finding $GH$: Given the centroid property, $GH$ is one-third of $BE$. Therefore: $GH = \frac{1}{3} \times BE = \frac{1}{3} \times 42 \, \text{cm} = 14 \, \text{cm}.$

Thus, the length of $GH$ is 14 cm.

Would you like further clarification or additional details on any of these steps?

Here are some related questions to expand your understanding:

1. How do we know that $G$ is the centroid of the triangle?
2. What other properties can we derive from knowing the midpoints of a triangle's sides?
3. Can the lengths of other segments, like $AM$ or $MN$, be calculated in this scenario?
4. How does the centroid divide the medians in a triangle?
5. What role does symmetry play in solving triangle midpoint problems like this?

Tip: In triangle geometry, the centroid (where medians intersect) divides each median into a 2:1 ratio, with the longer part being closer to the vertex. This is a useful property to remember when dealing with triangle medians.