Butterfly graphics mostly involve "Asymmetric Vieta Theorem", which is a frequently tested pattern recently. Of course, symmetry is not a problem, and asymmetry is not a pain point. As long as you master the routine, the problem can be solved.

Asymmetric Vieta theorem can be realized by "sum product relation" or "half generation". Of course, it is not as good as "structural symmetry". Method 1 is to use the sum-product relationship, method 2 is to construct symmetry, and the rest is left to you to explore.

I don't know what is the core of solving the conic problem. But it is hard to say that "Vieta Theorem" does not play a key role, after all, the ultimate goal or conclusion will be transformed here. Symmetry produces beauty, but reality is cruel, and asymmetry accounts for the majority. Therefore, plastic surgery has become in full swing, constructing asymmetry into symmetry, thus eliminating inner pain.

"Three points are collinear and construct a dual formula", which is the essence of Law 3. Method 3 has a more appropriate name-set point. Without special training, the process will be less noticeable. Actually, me, too. You don't have to refuse anything that doesn't look good. It's also a good choice to browse.

Set point is more widely used in parabola, because parabola only contains a square term, and elimination becomes simple. Setting the solution point avoids the asymmetry of Vieta's theorem, and naturally there is no need for those elimination techniques.

It is worth mentioning that the previous Vieta Theorem is generally a formula about slope (or intercept), and the solution point is directly converted into coordinates, which is essentially the same. However, in the face of asymmetric forms, the advantages of setting up points are truly reflected. I quite like this routine.