How to find upper envelopes of intersected lines in O(nlogn)?

Disclaimer: Yes, this is a homework and I am thinking about it for a couple of days but couldn't find a way to go.

So there are n straight lines (y= ax + b) and I want to find upper envelopes of them (bold part in the picture). It has to be in O(nlogn).

What I understand is, I need to find a way to ignore some of the lines, because if I search all of the lines it won't be O(nlogn).

I am thinking about a divide & conquer approach so that I can divide the list into two and recursively continue until the solution. But then I don't know how to get rid of some of the lines. Clearly, I don't need to consider some of the bottom lines in the picture because it's impossible for them to contribute to the solution.. But nothing came to my mind. Any hints are appreciated.

First we construct two different binary search trees for the lines, one with the lines sorted according to their a and the other according to their b.

Now we start considering the lines lmin, lmax, that are the lines with the smallest and the biggest a; they will contribute for sure with the points given from the intersections with the second smallest and the second biggest lines and we add to those 2 points to the upper envelope.

Now we consider the intersection (xi,yi) between lmin and lmax and we ideally draw the vertical x = xi line. We have now to identify the lines which intersect x = xi in a coordinate y0 s.t. y0 <= yi and remove all this lines from both the trees.

How can we identify these lines? We find all the lines with a b s.t. lmin <= b <= lmax, using the second tree.

At the end we'll also remove lmin and lmax from the trees.

Now we'll recurse on the new trees obtained.