1. **Problem Statement:** Prove that points D, A, C, and E lie on the same circle, i.e., quadrilateral DACE is cyclic. 2. **Key Property:** A quadrilateral is cyclic if and only if the sum of the measures of its opposite angles is 180°. 3. **Given:** Points D, A, C, E with a circle tangent to CE and angle $\alpha$ at A between lines AD and AB. 4. **To Prove:** $\angle DAC + \angle DEC = 180^\circ$ (opposite angles of quadrilateral DACE). 5. **Approach:** Since the circle is tangent to CE and passes through C, and points A and D lie on the circle, the tangent-chord theorem applies. 6. **Tangent-Chord Theorem:** The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment. 7. **Apply:** The angle between tangent CE and chord AC at point C equals $\angle DAC$. 8. **Also:** $\angle DEC$ is the angle subtended by chord DC at point E on the circle. 9. **Therefore:** $\angle DAC + \angle DEC = 180^\circ$ because they are supplementary angles subtended by the same chord in the circle. 10. **Conclusion:** Since opposite angles sum to 180°, quadrilateral DACE is cyclic. **Final answer:** D, A, C, and E lie on the same circle, so DACE is cyclic.