1. **State the problem:** We need to find the radius of a circle with diameter $AD$ given that $C$ lies on the circle, $B$ lies on $AC$, $E$ lies on $AD$, and $BE$ is parallel to $CD$. We know $AB=21$, $CD=18$, and $BE=13.5$. 2. **Recall the properties:** Since $AD$ is the diameter, the circle's radius is $r=\frac{AD}{2}$. Point $C$ lies on the circle, so $AC + CD = AD$. 3. **Use parallel lines:** Since $BE \parallel CD$, triangles $ABE$ and $ACD$ are similar by AA similarity. 4. **Set up ratios from similarity:** $$\frac{AB}{AC} = \frac{BE}{CD}$$ Substitute known values: $$\frac{21}{AC} = \frac{13.5}{18}$$ 5. **Solve for $AC$:** $$\frac{21}{AC} = \frac{13.5}{18} = 0.75$$ $$AC = \frac{21}{0.75} = 28$$ 6. **Find $AD$:** Since $AD = AC + CD$, $$AD = 28 + 18 = 46$$ 7. **Calculate radius:** $$r = \frac{AD}{2} = \frac{46}{2} = 23$$ **Final answer:** The radius of the circle is $23$ cm.