Based on the image provided, we need to calculate the measures of specific line segments (CD, CE, HD, GD, HG, and HF) using the fact that $H$ is the circumcenter of $\triangle BCD$.

Key Information:

• $H$ being the circumcenter implies that it is equidistant from all vertices of $\triangle BCD$.
• Thus: $HB = HC = HD$
• Additionally, the given values of side lengths or other relationships should help us compute the required lengths.

Let's solve:

1. $CD$: Since $CD$ is the side of $\triangle BCD$, it can be determined using the given lengths and triangle relationships.

2. $CE$: Segment $CE$ can be calculated from the diagram and given lengths. Check for any specific relationships $CE$ might have with $H$ or other segments.

3. $HD$: Using the property of the circumcenter, $HD$ = $HC$ = $HB$.

4. $GD$: Segment $GD$ may relate to the perpendicular distance or radius from $H$ to $D$.

5. $HG$: Check the lengths relative to the position of $H$ in the diagram.

6. $HF$: $HF$ could be derived based on relationships within the geometry.

If you want step-by-step calculations, confirm the exact dimensions or relationships visible in the diagram. Would you like detailed explanations for each calculation?

Relative Questions to Expand Understanding:

1. How is the circumcenter's property of equidistance helpful in solving triangle geometry problems?
2. What is the role of perpendicular bisectors in identifying a circumcenter?
3. How can the lengths of the circumradius help deduce unknown side lengths in a triangle?
4. How do we apply the Pythagorean theorem in determining distances within a triangle diagram?
5. Why is it important to understand circumcenter properties in relation to practical applications like navigation or construction?

Tip: Always verify the relationships and symmetries when working with geometric centers to ensure accurate calculations.