Final answer:
Apollo indirectly influenced the fates of Achilles and Neoptolemus in Greek mythology, with Apollo guiding the arrow that killed Achilles and with Neoptolemus avenging his father's death.
Step-by-step explanation:
The relationship between Apollo, Achilles, and his family, particularly Neoptolemus (also known as Pyrrhus), is driven by the narratives of the Trojan War, as portrayed in ancient Greek mythology and literature. Apollo, being a god, indirectly influenced the fate of Achilles and Neoptolemus. Achilles, although a great hero of the Greeks and son of the nymph Thetis and Peleus, was mortally wounded by an arrow guided by Apollo, which struck his only vulnerable spot, his heel. Following Achilles' death, his armor was highly coveted, leading to a grievous dispute between the heroes Ajax and Odysseus.
Neoptolemus, the son of Achilles, also figures into the mythology as a warrior who avenged his father's death during the fall of Troy. Some sources tell us that Achilles was worshipped as a near-divine hero after his death, with Apollo playing a part in his legend. The connection between these figures underscores the importance of destiny and divine intervention in ancient Greek stories.