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Similar to point location
I have a following problem:
Given a set $S$ consisting $N$ triangles (possibly overlapping). Answer queries (online) of the form: given a point $P$ is there a triangle $T$ in $S$, such that $P$ lies inside $T$?
I am interested in solution (data structure) which:
$*$ Answer query faster than $O(N)$ (worst case)
$*$ Preprocesses triangles faster that $O(N^2)$ (worst case)
Problem looks classic, but I am new to computational geometry.