in fact, in point of fact, as a matter of fact
Solution 1:
The three expression are actually synonyms with different registers. As a matter of fact and in point of fact sound more formal and as Ngram shows they are by far less common than the more popular and colloquial in fact. It appears that only as a matter of fact has a specific etymology.
As a matter of fact:
Etymology: (from Etymonline)
- also matter of fact, 1570s as a noun, originally a legal term (translating Latin res facti), "that portion of an enquiry concerned with the truth or falsehood of alleged facts," opposed to matter of law. As an adjective from 1712.
Meaning:
actually in (point of) fact I did vote for her, as a matter of fact.
Usage notes: used to emphasize the truth of what you are saying. ( from Cambridge Dictionary of American Idioms)
In point of fact:
- in reality or actuality; "in fact, it was a wonder anyone survived"; "painters who are in fact anything but unsophisticated"; "as a matter of fact, he is several inches taller than his father"
In fact:
- in reality or actuality; "in fact, it was a wonder anyone survived"; "painters who are in fact anything but unsophisticated"; "as a matter of fact, he is several inches taller than his father"
As a side note, I personally tend to use the expression as a matter of fact in formal business conversations.
( source: TFD) .