I think it all comes down to what the writer/speaker thinks about the reader/listener
ie. If it is new to the reader use 'a', otherwise use 'the'
as for your examples
So I just cannot see the difference between "this is a book I told you about yesterday" and "a woman who fell 10 meters from High Peak was lifted to safety by a helicopter."
Why can a definite article be used with "woman" for introducing for the first time and not with "book"?
If you told me about the book yesterday, then it wouldnt be new information so it would be: "this is the book i told you about yesterday", Using 'a' in this case almost implies that I forgot about it, and you are reminding me
For the second one, I have never heard of the[1] woman (in the article) before so you describe her as 'a' woman and not 'the' woman.
the[1] last examples:
It should be "the" according to the explanation people have given me, right? Unless there is more than one letter he sent to Burnham on June 20, which is unlikely.
This does read to me as if there is more than one or that he wrote so many letters over the years that they can only specify which one they are talking about by date.
Likewise, it should be "the" unless Burnham wrote several letters that contained a description of the meeting.
it is 'a' because he wrote only one letter. "Burnham wrote one letter to Olmsted...", if you said 'the' in this case, it would imply that I knew about the letter before you said the sentence
[1] I use 'the' because you know what I'm refering to here
"Few" and "a few" are significantly different in grammar and meaning.
"A few" is neutral: it simply refers to a small number of whatever, without suggesting any expectations.
"Few" triggers negative polarity expressions (such as "any" and "yet"), and emphasises the smallness of the number, and suggests that a larger number might have been expected.
A few people have registered.
A few people have an idea what's going on.
Contrast this with:
Few people have registered yet.
Few people have any idea what's going on.
both of which are emphasising that the number of people is small.
A few with negative polarity items is not grammatical:
*A few people have registered yet.
*A few people have any idea what's going on.
There is no real answer to your question: like most "why" or "what is the reason" questions about language, the only answer is "because that's the way it is".
Best Answer
This is an interesting construct and a great question.
Let's start by explaining it this way: Instead of using the proper name, use a generic word instead. Then the sentences will make a lot more sense:
Those probably make sense, and articles do not seem odd or out-of-place.
When you substitute the person's name for the generic word, the sentence carries more meaning.
If I heard this, I would assume:
Therefore, I'd probably answer with something like:
or:
As for:
That pretty much says the same thing as
but it has a more poetic ring to it; you are ascribing a state of astonishment to John.
As for the other example in your comment:
This is different; in this case, I would interpret the definite article to mean that Sara Hope is a person with some fame or celebrity, and Mike was with the famous celebrity, not some unknown person who happened to have the same name.
At any rate, these constructs are grammatical and valid, but they are also a bit unusual. I would caution against overusing them in casual conversations.