1. **State the problem:** We are given triangle BCD with circumcenter H inside it. Known lengths are EC = 7, HF = 3, HB = 13, and BD = 24. We need to find the missing segment lengths EH, FD, and CD. 2. **Recall properties of a circumcenter:** The circumcenter is equidistant from all vertices of the triangle. This means: $$HB = HC = HD$$ Given $HB = 13$, we have $HC = 13$ and $HD = 13$. 3. **Use given segments and points:** Since $BD = 24$ and $G$ lies on $BD$ with $GD = 12$, then $BG = BD - GD = 24 - 12 = 12$. 4. **Find CD:** Since $CD$ is a side of the triangle and $HC = 13$, $HD = 13$, and $H$ is the circumcenter, $H$ lies on the perpendicular bisector of $CD$. The length $CD$ can be found using the triangle properties or coordinate geometry, but since $HC = HD = 13$, and $CD$ is a chord of the circumcircle, $CD$ is less than or equal to $2 imes 13 = 26$. Without additional data, we cannot find $CD$ exactly here, so we keep it as unknown. 5. **Find EH:** Point $E$ lies on the circle with center $H$ and radius $13$ (since $HB=13$). Given $EC=7$, and $H$ is the circumcenter, $EH$ can be found using the triangle formed by points $E$, $H$, and $C$. 6. **Find FD:** Given $HF=3$, and $F$ lies on the triangle, we can use the properties of the circumcenter and distances to find $FD$. **Summary:** Without additional coordinates or angles, the exact values of $EH$, $FD$, and $CD$ cannot be determined from the given data alone. More information is needed to solve for these segments. **Final answer:** - $EH$: Cannot determine with given data - $FD$: Cannot determine with given data - $CD$: Cannot determine with given data