I saw that in most go games the opponents started by placing chips 3 vertices away from the corners. Both players started fighting for the centre only after half of the game had passed. Is there any particular reasoning for this?
Common starting practices are to open by placing stones on a Hoshi (4-4 point), or close to it, in order to "claim" that corner. As the players are trying to secure as much area as they can, it makes sense to start out in a place that is easily defended, and that is something that's true for corners more than for any other location on the board at the start of the game.
Taking control of the center is much easier if the player has a solid base or even a number of solitary stones already placed, which obviously isn't the case at the start of the game. Furthermore, by trying to gain control of the center, your opponent would likely take all or most of the corners for themselves, which will eventually result in having to defend the center against all flanks.
In a real game, of course none of this will be this clearly separated, but those are the essentials it comes down to. Generally, you're not looking to immediately engage with your opponent, but rather to define areas of advantage for yourself first.
Because this question, just like most questions about Go, pose a larger complexity than would be feasible to go into here, I'll leave it at that and provide a few links that are likely to be reachable in the future to read further:
It should be noted that this is true for 19x19 boards - on 9x9 boards, you will much more commonly see players starting out at the Tengen (center starpoint), or at the very least try to compete for it much earlier on, simply because there isn't as much area to gain in the corners by comparison, and approaching the same strategy as on a 19x19 board would have you run into your opponent much faster than you'd like.
Capturing territory around the edges of the board is simpler and quicker, because the edge forms a boundary which does not have to be constructed with stones, and which can not be attacked from "outside".
Once a player has a safe territory attached to the edge, it can be extended towards the center of the board.
It is also relevant that a border just three stones wide around the entire edge of the board contains more than half the total board area. Of course a real game situation will be more complex than that simple piece of mathematics, but the general principle it demonstrates is important.
Because you need less stones in the corner to live than in the center. And by extension, you can surround more area in the corner with the same number of stones.
Some figures to visualize this:
Live group in the corner, 6 stones:
+----- |.xx |x.x |xx |
Live group in the middle, 10 stones:
xx x.x xx.x xxx
(See Smallest Group with Two Eyes on Sensei's Library.)
Those 10 stones in the corner could cover much more points:
+------- |....x |.x..x |....x |....x |xxxxx |
It is plain and simple that it is much easier and more efficient to make life groups and territory in the corners than along the edges or in the centre. In the corner, you need two outer walls, along the edge three, and in the center four to enclose your territory.
And Go is about enclosing territory, after all.