Soft question- The Bashing Technique and Other powerful techniques for Olympiads
"Bashing" is a term for brute force methods, applied with very little cleverness. These are looked down on in contest mathematics, both because they aren't "pretty" and because they tend to take more time and computational effort than is practical in a live contest.
How do you tell if a solution is "bashing"? That's an entirely subjective judgment. The more you like it, the less likely you are to call it bashing.
What's the advantage of methods that might get called bashing? Reliability. Often, that method is something you know will work if you put enough time and effort into it. For example, consider the two top-rated answers to this recently active inequality problem; one is a short and sweet application of a classical inequality to eliminate the square roots, while the other is a very long slog of multivariable calculus and numerical root-finding, finding all twenty critical points of a function in order to find its minimum. The latter is certainly fairly bashy - I might not have gone through with it if I had realized how much work it was from the start - but it's also a complete solution, where the former isn't. After the simplification, that attempt stalls out with no clear next step.
I think in general the term bashing is used to describe an approach which is about applying a small set of manipulations over and over until you get an answer. These approaches usually don't require much insight.
One other trait of bashing which I can think of is that it is often very computational. This will usually go hand in hand with bashing's non-insightfulness.
The example you gave of using coordinate geometry to solve geo problems is a prime example of bashing as it is just a bunch of messy algebra after you translate the terms of the problem to coordinate geometry.
In inequalities examples of bashing they are:
Buffalo Way. See here: https://artofproblemsolving.com/community/c6h522084
uvw . See here: https://artofproblemsolving.com/community/c6h278791
Bashing is usually a way of using brute force to answer a question. In most cases, it doesn't require any mathematical knowledge of theorems and techniques.
The way to go in basing is usually just trial and error but for higher efficiency, you can use theorems to set upper and lower bounds for your answer. This will increase efficiency while bashing.
The most common bashing is done with induction. Where you are trying to prove a theorem so you substitute 1 and then 2 and then we know that if it satisfies for n, it should satisfy for n+1
I must warn you that in olympiads like RMO, INMO, EGMO, USMO, AIME, IMO, etc, Basing is looked down upon and will have less weightage because in these olympiads they check your method of answering as well as your answer.