1. **Problem statement:** We need to find the measure of angle $\angle BAC$ in a circle where $B$ is the center, and $A$ and $C$ lie on the circumference. 2. **Key fact:** The angle subtended by a chord at the center of the circle is twice the angle subtended at any point on the circumference on the same side of the chord. 3. Given that $B$ is the center, $\angle BAC$ is an inscribed angle subtended by chord $AC$. 4. The angle at the center $\angle ABC$ is twice $\angle BAC$. 5. From the problem, the angle between line $CE$ and chord $AC$ is $52^\circ$. Since $E$ lies below $C$ and $D$ lies above $C$, and $B$ is the center, the central angle $\angle ABC$ corresponds to $104^\circ$ (twice $52^\circ$). 6. Using the relationship: $$\angle ABC = 2 \times \angle BAC$$ 7. Substitute the known value: $$104^\circ = 2 \times \angle BAC$$ 8. Solve for $\angle BAC$: $$\angle BAC = \frac{104^\circ}{2}$$ $$\angle BAC = 52^\circ$$ **Final answer:** $\boxed{52^\circ}$