B+2P vs 2P - draw or win?

I haven't studied engames much, so I'd like a bit of help with this one. I reached this position and my opponent offered me a draw. I declined and won the game, but my opponent made some mistakes, so I'm still not sure if this position is drawn or not:

It does look like draw to me. I have wrong bishop, it can be only used for protecting my own pawns and restricting his king. My plan was to try to capture his pawn on a-file and then protect my a-pawn with bishop while going with my king for his h-pawn, or vice versa. He can't be at both sides of board at once, which is typical element of endgames. The thing is how to restrict his king from protecting his a-pawn. I only managed to do that because of some of his inaccuracies.

who is on move?

White, like it says in diagram. He played Ke4 here, which is good move in my opinion. I can't get to his h-pawn and he gets closer to his a-pawn than me.

Coordinates should be reversed, I'm sorry about that, will change it now.

black will win

What about 2...Kg6 3.Ke5? I think nothing has changed if white plays 3.Ke5.

Won for Black, of course.

Thanks! What is the technique for such endgame? I know it's something obvious, probably the same thing I mentioned in first post, but I'd just like to be sure I played this endgame with correct plan.

As I imagined. Thanks for explanation, I thought there is a lot of complexity when you have the wrong bishop, but position seems quite simple (and won) now.

White wins

It really took you 14 years to work this out? Impressive dedication.

And now it will take him another 14 years to work it out for the black bishop.

White wins

The extra Bishop wins you opposition 100% of the time. Only time the extra Bishop doesn't win is when stalemate cages are created, like the following case:

This is a draw no matter who is to move. You cannot get Black out of the corner. He will toggle Kh8-g8-h8-g8 until you cover g8 with the Bishop or King and stalemate Black, or go 50 moves.

Also, saccing the bishop wont do anything

..

This position is indeed a draw, but if you put the bishop on a dark square white wins, as he can sac the bishop and win the pawn ending.

Sure about that? That particular position looks drawn even if you put the bishop on a dark square.

That is true

A piece up... nvr draw.

that’s obviously not true