It is done using symbolic algebra, not just assigning specific coordinates if that's what you're asking.
Let's say you're given three parallel lines. You can put your x axis along the first line. Then its equation is y = 0. The other two lines necessarily have equations y = a and y = b for some reals a and b. Then you calculate the other quantities involved via a and b and other parameters you have to introduce. At the end you calculate the coordinates of your three points and verify, symbolically, that they lie on the same line.
Let's say you're given three parallel lines. You can put your x axis along the first line. Then its equation is y = 0. The other two lines necessarily have equations y = a and y = b for some reals a and b. Then you calculate the other quantities involved via a and b and other parameters you have to introduce. At the end you calculate the coordinates of your three points and verify, symbolically, that they lie on the same line.