Question: I don't know if this is the right place to post this or not, but I will go ahead anyway (sorry if it ain't the right place)

Yesterday I was discussing a particular theorem of geometry with my brother which he just learnt in the school. I had asked him if he knew the proof for it, he replied saying his teacher has said that wouldn't be necessary.

Then, I asked him to sit and try to prove the theorem. He said that knowing the theorem counts for more than knowing the proof. How do I explain to him that knowing the proof is more important and how it can even help expand his thinking?

I know this question doesn't have a single pointed answer as is pre-requisite for questions posted here, but I would appreciate any replies

Answer: 
I think the dislike for proofs, and the belief that they must not be important, comes from two misconceptions:

1. That proofs are all two-column proofs.
2. We prove things to find out if they are true.

The first is blatantly false. Most proofs are paragraph proofs, and don't have to show every tiny little step. No one will be displeased with you if you commute a few variables here and there without explicitly saying so. Often, proofs are taught in high school with a very specific format, and straying from it is penalized. This makes it seem pretty arbitrary and dumb.

The second is a little more subtle. Proofs do, in fact, prove that things are true. One benefit of that is that we can be absolutely certain we're working "on solid ground", so to speak. But if you aren't sure if something is true, you don't go trying to prove it first. The more important aspect of a proof is that it is a justification as to *why* something is true. Proving the "why" of something gives you

* the potential for more generalization
* a larger toolbox for proving other things

For the first point, if you notice Möbius inversion works for the totient function, you could just prove that and call it a day. But if you look at the proof, you might notice that it doesn't actually involve computing the totient function at all, and it just uses the fact that it's multiplicative. (Well, more likely it'd take a bit of needling to give up that fact, but it's still possible to pry it out) This gives you a far more general theorem, one that would be much harder to find if you only knew the special cases, without proof.

For the second point, anorton's answer has a very good example involving double and triple angles.

Ultimately, proofs are for verifying results that you're pretty sure are true. Because everything you're taught has been proved, this seems useless for quite some time. But when you actually *do* need to find new things, proofs are indispensable.

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The other comparison I've heard is to science labs. Yeah, why go and *test* that $F = ma$ is true if we can just check the physics book? Because that's not how science is done out in the real world. You use it to support hypotheses that you have. (Although in math, you get to *prove* them!)

On the other hand, this can be a dangerous analogy, because this is exactly the opposite of how math works. You don't conjecture things on paper and prove them in the lab, you conjecture with numerical examples (the lab) and then you prove things on paper.
One approach would be to give an example of a theorem that sounds so absurd that noone in his right mind would accept it without proof. For example, mention the Banach-Tarski result to the effect that the unit ball can be decomposed into 5 parts which can then be reassembled by rigid motions into... two unit balls.

Another example would be the equality of $0.999\ldots$ to $1$. Of course, this one he won't believe even after you give him a proof :-)