these three points are on one line
these three lines intersect at one point
these two lines are parallel
these four points are on a circle
these two angles are equal
etc ...
There are things that absolutely can be done via coordinates, if one can brute force large algebraic expressions.
If memory serves right, there's a certain subset of geometry that decidable in that way. Ie you can just run an algorithm.
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.
Yes.