deletions | additions       diff --git a/Line_Moving_Queries_We_represent__.md b/Line_Moving_Queries_We_represent__.md  index 2151f60..5fc9e99 100644  --- a/Line_Moving_Queries_We_represent__.md  +++ b/Line_Moving_Queries_We_represent__.md 
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# Line-Moving Queries  We represent the problem in the dual-plane. We find that by this representation our task is to find the closest  line \(s^*\)furthest  below our query point \(l^*\). We solve can find  thisproblem  by first finding all \(O(n^2)\) intersections constructing the arrangement  of our lines. We then finding lines \(S^*\) in  the convex hull dual-plane, and then building a trapezoidal-decomposition on top  of these intersections \(O(n^2 \log n)\) time, \(O(n^2)\) space. this. This allows us to find the face that any point we query resides in in \(O(\log n\)\).  We then perform a vertical line-query from \(l^*\) and find the bottom-most intersection point. point with the convex face.  We can map this back to the line \(\in S*\) that we're after. ## Dual Problem