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: 
In my opinion, they're not important. But let me qualify that in a massive way.

Suppose we're interested in solving some particular real-life problem. We're going to make some "let" statements. Like maybe we've got $(x,y) \in \mathbb{R} \times \mathbb{R}$. And to solve the problem, we're going to let $(r,\theta)$ satisfy $x = r\cos \theta$ and $y = r\sin \theta$. Why are we allowed to do this? We're actually leveraging a theorem:

> 
> For all $x,y \in \mathbb{R}$, there exists $r \in \mathbb{R}\_+$ and $\theta \in \mathbb{R}$ such that $x = r\cos \theta$ and $y = r\sin \theta$.
> 
> 
> 

This legitimizes the "Let" statement. So theorems tell us the rules of the game; and therefore, the range of problems the human race is capable of solving is hard-limited by the theorems we know. So theorems are important.

Does that mean you *personally* ought to know the proof of a theorem? That we're not allowed to use it 'til we know the proof? Of course not.

That being said, if you want the ability to prove your *own* theorems and thereby move the human race forward, expanding the range of problems that we're capable of solving, well you'd better start going beyond the question: "What can I do with this theorem?"

You'd better start asking: "How do we even know its true?"

By the way, most theorems in mathematics aren't proved because of their direct applications to real world problems. This begs that we ask: "Why should we care about theorems that don't have direct applications to real-life problems?" You should post another question if this topic interests you.
Proofs are important because they tell you why the theorem is true: no amount of practical experience with the use of some helpful formula can ever explain its existence, and this renders mathematics just a bland narrative of assertions to be judged on the merit of their applicability to one's occasional interests. To take a geometric example, I don't think it's at all relevant to my life that the [incircle](http://en.wikipedia.org/wiki/Incircle) of a triangle is centered on the intersection of the three angle bisectors, but after a moment's thought, I found a cute proof by symmetry; you wouldn't think that would work, given that the triangle likely is completely asymmetric. So now I like the theorem, and ten minutes ago, I didn't care. I still have no use for it.

**Edit:** The proof, by the way, is to focus on the circle rather than the triangle. Pick any two tangent lines and let $P$ be their point of intersection. The line from $P$ to the circle's center is an axis of symmetry for this picture, which you can prove by dropping radii to the points of tangency and using similar right triangles. Therefore it bisects their angle, and that means that the center lies on the angle bisector between any two sides of a triangle to which it is tangent. If that's not Euclid's proof, it certainly could be.