Regarding (1), I want to say that according to my own ideas, the point method is helpful to express some formulas and finally simplify the relationship between what is sought and what. For example, for the problem you mentioned, we can set the abscissa of point M as m, so that point M on the parabola must satisfy the analytical formula of parabola, and then the ordinate of point M is an algebraic formula about m, so you can express the triangle area you need through the area relationship between several triangles, and then find that the relationship between the triangle area you need and m is a quadratic function, which requires you to take the maximum, so if you simplify it, there must be a maximum in the end. Of course, this problem is still difficult, and it is necessary to discuss the value and range of m, and finally get a simplified result.

In fact, to put it bluntly, it is an auxiliary element to help solve problems. Why use it? Because it's hard to find the answer without using it. We must use some simple things we know to express some abstract things we don't know.

(2) Let me make an analogy. There is a little m on the parabola y =-x 2+3x+ 1. Where the point M moves, the distance to the straight line y=x is the longest, so you can set M(m,-m 2+3m+1) and get the point M through a series of exercises.