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A player on the opposing team kicks the ball and the ball crosses the goal line. However, before the ball touches either the net or the ground inside the goal, the goalkeeper catches it. Does the ball still count as a scored goal?

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Yes.

From page 34, 2014/15 FIFA Laws of the Game:

A goal is scored when the whole of the ball passes over the goal line, between the goalposts and under the crossbar, provided that no infringement of the Laws of the Game has been committed previously by the team scoring the goal.

In the game of association football, the ball is always out of play when it crosses the line either in the air or on the ground. It never has to hit the ground to be out, like in other forms of football, such as rugby. From page 33, 2014/15 FIFA Laws of the Game:

The ball is out of play when:

  • it has wholly crossed the goal line or touch line whether on the ground or in the air

Nets only exist to make it clear to the refereeing team that the ball passed under the crossbar and between the post. They're not even required in the Laws, although most competitions will require them in their competition rules to make things easier for the refereeing team. From page 11, 2014/15 FIFA Laws of the Game:

Nets may be attached to the goals and the ground behind the goal, provided that they are properly supported and do not interfere with the goalkeeper.

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    For clarity, it could be added that "wholly crossed" does not simply mean that you can see green grass between the ball and the goal line (as that is not necessarily a goal). To understand when a ball has crossed the line, one can imagine the ball as a cube, rather than a ball. The whole cube must be over the line in order for it to be counted as a goal. +1 for a great answer though! – Qvist Jul 23 '14 at 17:46
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    The cube analogy is good, but it only works if the cube always has a face that is parallel to the ground. Otherwise, when the cube has opposite vertices perpendicular to the line, the measurement will be out by ~73% (1-sqrt(3) due to Pythagoras' theorem). An illustration that requires less care is imagining that the touch line extends infinitely up to form an infinite rectangular prism and isn't just marked flat on the ground. If any part of the ball is in this infinite prism, it is still in. – studro Jul 24 '14 at 0:50
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    Your analogy is good as well, and I do not have any criticism of it! As for the cube analogy, of course the "mental" cube has to be drawn with one face parallel to ground - and also parallel to the goal line. However, the cube shouldn't be imagined as something that rotates at all. Rather, it must always stay firm. The ball on the other hand, which is (almost completely) spherical can rotate all it wants, since it does so inside the cube. Really, the sphere would meet the cube in exactly six spots, so it would create a little friction - but that tiny detail must be ignored ;) – Qvist Jul 24 '14 at 5:16
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Yes this counts as a goal. Once the entire ball passes the goal line, it's considered as a goal, no matter it is on the ground or in the air. However, this is very tough for the referee and his assistants to see and many times such goals are not awarded. That is why the goal line technology was introduced, to reduce controversy in such cases

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