An apple a day keeps the doctor away.
Curiosity killed the cat.
I try to learn the articles but I can't get these cases. Why should say the doctor instead of a doctor and same is about the cat?
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Let's say you are in the emergency room of a large metropolitan hospital where there are many physicians on duty. You can say
I am waiting to be seen by a doctor.
I am waiting to be seen by the doctor.
The choice of article reveals how you are thinking of it:
a doctor = one of the many physicians who are here
the doctor = the physician (whoever it turns out to be) who will be assigned to my case, the one who comes into examination room where I am waiting.
In much the same way, you can say:
An apple a day keeps the doctor away.
Which doctor? No doctor in particular, but the one you will have to see if you should become ill.
Now, let's say there is a fable about a fox who wants to eat some grapes. We can refer to the fable as
The story of the fox and the grapes
Which fox? The fox who figures in the fable. The fox who wanted to eat some grapes.
Which grapes? The grapes which figure in the fable. The grapes the fox wanted to eat but could not reach.
In much the same way, we can say
Curiosity killed the cat.
Which cat? The cat in the proverbial example that we should bear in mind.
