What is the most likely source of profuse bleeding when extracting a maxillary third molar that is displaced into the sinus?

The most likely source of profuse bleeding when extracting a maxillary third molar that is displaced into the sinus is the posterior superior alveolar artery. This artery is a branch of the maxillary artery and supplies the maxilla, including the maxillary molars and the maxillary sinus. During the extraction of a maxillary third molar, especially when there is a complication such as displacement into the sinus, the posterior superior alveolar artery is at risk of being compromised or injured, leading to significant bleeding.

When a tooth is located near the sinus, the risk of damage to surrounding vasculature increases, particularly the posterior superior alveolar artery, which courses very close to the roots of the maxillary molars. If this artery is severed during extraction, it can result in substantial and profuse hemorrhage.

In this context, while the maxillary artery, facial artery, and inferior alveolar artery are also branches of the external carotid system and have their own anatomical relationships, they do not directly supply the area involved in a maxillary third molar extraction as critically as the posterior superior alveolar artery does. Therefore, the greatest risk for bleeding comes from this specific artery during such a procedure.