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this is not proof, though. can you prove it in general?
Welcome to the forum.
In mathematics, you don't understand things. You just get used to them.
I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.
You must be careful, but many proofs can be made by pictures alone. You must use caution because picture proofs can charm your intuition which can be easily fooled and furthermore, it is easy to ignore important conditions to your theorems!