Case 1: Antonia's two darts are closer to the center than Beatriz's. Antonia wins.

Case 2. Antonia's two darts are further from the center than Beatriz's. Beatriz wins

Case 3: The first dart Antonia throws is the one closest to the center, followed by Beatriz's, and farther, Antonia's other dart → Antonia wins.

Case 4. The first dart Antonia throws is the one farthest from the center, followed by Beatriz's, and closer, Antonia's other dart. Antonia wins.