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He is a friend of my father.

He is a friend of my father's.

He is my father's friend.

Are all the sentences correct? If so, what are the differences?
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They are all correct.

The second is more usual than the first. The first and second are nearly identical in meaning.

While the first two suggest that he is one of my father's (possibly many) friends, the third suggests that he is the one particular friend that we were just talking about.

a my friend and the my friend are grammatically impossible in English.
They are usually rendered as a friend of mine and my friend, respectively.
These correspond to expressions like a friend of my father's and my father's friend.

CJ
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CalifJim .... the third suggests that he is the one particular friend that we were just talking about.
Also, in many circumstances,
He is my father's friend.
is taken to mean:
He is my father's only friend.
(my father has only one friend)

However, strictly speaking, only would be necessary for this meaning.
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Comments  
Hi

My father's friend.

There is the possessive 's inflection with the possessor: father.

We can not put another deteterminer between the determiner (my) and the inflected noun (father's) to express we are referring to, say, two of them.

However, 'My father's two friends' is acceptable, although rare.

We prefer to use another grammatical structure:
two friends of my father's

In cases where we refer to family relationships, we may drop the inflection, 's.
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