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accurately investigated
"carefully researched." Luke was careful to find out exactly what happened. He probably talked to the different people who saw what happened to make sure that what he wrote down about these events was correct.
most excellent Theophilus
Luke said this to show honor and respect for Theophilus. This may mean that Theophilus was an important government official. This section should use the style that your culture uses to address people of high status. Some people may prefer to also put this greeting at the beginning and say, "To ... Theophilus" or "Dear ... Theophilus."
most excellent
"honorable" or "noble"
Theophilus
This name means "friend of God." It may describe this man's character or it may have been his actual name. Most translations have it as a name. (See: rc://en/ta/man/translate/translate-names)