Let p₁, ..., p_n be points of polygon in clockwise order.

Now you're given point q and have to check whether it lies in polygon. At first check that it lies in angle formed by points p₂, p₁, p_n. If that's false, then answer for sure is no. Otherwise let's notice that function "f(i) =  is it true that triplet (p₁, p_i, q) is right-orientated" is monotone (i.e. it's always true at first, then always false). Do binary search on i checking the condition using orientated triangle area. After you've found the first i such that triplet (p₁, p_i, q) is left-orientated, check that q is in triangle p₁, p_i - 1, i. Done!