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In 1979, the NBA voted to have a 1-year trial of the 3-point line, which later became permanent. Although the ABA used a 25 foot 3-point line, the NBA used a 23.75 foot line. How was the 23.75 number settled on? Was it random?

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While I can't find any sources that clearly specify why the line was set at that level, one can speculate on a few good reasons for the distance being what it is based on deduction.

Firstly, a standard NBA court is 50 feet wide. In the original implementation of the 3-point line in the American Basketball League(1961-1962), the line was set at 25 feet, as you mentioned. When the ABA introduced the 3-pointer in 1967, they actually used the same 3-point line currently used in the NBA, which is 23 feet, 9 inches. (Source) When the ABA folded into the NBA, initially, the 3-point line was not included in the deal. (Source) It took the NBA 3 years to adopt the 3-point line and they simply used the same measurements that the ABA had used.

However, the line is drawn to allow 3 feet minimum along the sidelines to provide some space to shoot a 3-pointer from all legal angles. This means the distance is only 22 feet from the corners where the line is parallel to the sideline. For these reasons, it is likely that a 25 foot line would create a major advantage to shooting a three from the sideline areas in that it is 3 feet closer to the basket. As it is, there is still something to be said about the 1 foot 9 inch difference, but looking at an image of the current NBA court may help demonstrate another point of potential issue. Pushing the line back further would create additional space inside the 3-point line which would cause the defense to spread even further to defend outside shooters. The NBA attempted to increase scoring for 3 years starting in the 1994-1995 season by reducing the distance to a uniform 22 feet around the basket, with the change ultimately reverted in the 1997-1998 season. One could then infer that any longer distance would reduce scoring -- something that the NBA clearly doesn't want.

An additional point to be made is the comparison between the college and pro levels in the US. At the college level, the distance was initially set at 19 feet 9 inches during the 1987-1988 season -- a full 4 feet closer. It was later lengthened to 20 feet 9 inches in 2007. As the collegiate level is an amateur league, it makes sense that the court is shaped with that in mind. Using a shorter line enforces the relationship between amateur and professional.

Finally, using the measurements below -- it is likely that the measurement was set based on distance to the backboard, though it is usually referenced by distance to the middle of the hoop. The image below shows a 15 inch measurement between the center of the hoop and the backboard. Adding 15 inches to 23 feet 9 inches equals a full 25 feet, backboard to top of the 3-point line.

enter image description here

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