What is the difference between a coordinate geometry proof and a proof method that does not require coordinate geometry? when would it be appropriate to use a coordinate proof rather than another proof method?

A coordinate proof geometry is that which  uses a coordinate system. it f allows you to use algebraic techniques  . You can use coordinate proofs  when needed that you could do a synthetic proof. Sometimes it is very hard to do algebra  coordinate geometry helps you to use equations of lines, the midpoint formula, the distance formula, equations for circles, ellipses, etc...now take exampl you can prove the Pythagorean theorem using a coordinate proof: without loss of generality now  place the right angle of the triangle at the origin so that the the legs lying on the positive x and y axes.  now Let the vertex A be at (a,0), andthe vertex B be at (0,b) and the vertex C at (0,0) with  . Then by the distance formula u can solvehope it helps