Are ~는 중이다 and ~고 있다 the same?
I learned that both of these constructions translate to "I am (verb)ing". For example:

저는 먹고 있어요 = I am eating
저는 먹는 중이에요 = I am eating

Do these sentences mean exactly the same thing, or is there a subtle difference?

Question

Are ~는 중이다 and ~고 있다 the same?
I learned that both of these constructions translate to "I am (verb)ing". For example:

저는 먹고 있어요 = I am eating
저는 먹는 중이에요 = I am eating

Do these sentences mean exactly the same thing, or is there a subtle difference?

Answer

~는 중이다 and ~고 있다

They are synonyms, but there are cases where one is preferred over the other.

As I see it, ~는 중이다 is more explicitly "in the process/middle of -ing", which always suggests something ongoing.
-고 있다 on the other hand can also mean "in certain state/action" which may not mean actively going on.

Here are examples where the two make little difference.
- 나 지금 가는 중이야 = 나 지금 가고 있어. I am on my way there right now. 
- 저는 (밥을 / 저녁을) 먹는 중이에요 = 저는 먹고 있어요. I am eating.
- 지금 식사 중이야 = 지금 식사하고 있어.  (same)

These examples are about ongoing events, so ~ 중이다 is preferred.
- 1951년 한국은 전쟁 중이었다:  Korea was in war in 1951. (전쟁하고 있었다 sounds unnatural)
- 수업 중(in class), 가동 중(in operation), 방송 중(on air): common set noun phrases implying a process.

The following are not so much about an explicit process, so -고 있다 sounds more natural.
- 잘 지내고 있니? = 잘 지내니?  Is everything going well with you?
- 이 꼬마들(이) 지금 싸우고 있다 = 이 꼬마들 지금 싸운다: These kids are quarreling now.