The most common and most likely answer to this crossword clue is the 9 letter word EUCLIDEAN
I've seen this crossword clue in The Times
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More clues leading to the result EUCLIDEAN
Buddy explains!

The straight part
We know that "of ancient geometry" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case of ancient geometry = euclidean.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does EUCLIDEAN mean?
EUCLIDEAN adjective- Adhering to the principles of traditional geometry, in which parallel lines are equidistant.
- Of or relating to Euclid's Elements, especially to Euclidean geometry.
- Of or relating to Euclidean zoning.