Yes but if the first G is randomly chosen
G Is not randomly chosen. G is known as the generator point, and is the same for every private/public key pair in bitcoin. Specifically, it is:
Compressed: 02 79BE667E F9DCBBAC 55A06295 CE870B07 029BFCDB 2DCE28D9 59F2815B 16F81798
Or
Uncompressed: 04 79BE667E F9DCBBAC 55A06295 CE870B07 029BFCDB 2DCE28D9 59F2815B 16F81798 483ADA77 26A3C465 5DA4FBFC 0E1108A8 FD17B448 A6855419 9C47D08F FB10D4B8
The reason it can be compressed is because you don't have to store the y coordinate; it is sufficient to store the x coordinate and whether the y coordinate is positive or negative, and the y coordinate can be calculated.