Where the Wild Things Are

All submissions for this problem are available.
Read problems statements in Mandarin Chinese and Russian as well.
Recently, Max has visited the place where the Wild Things are again. He was very happy to know, that the monsters are living in peace now.
His friend Carol was so excited about Max's coming back, that he proposed everyone a brand new game. Unfortunately, all the Wild Things refused to play the game, except Max and K.W., so there were only three players playing the game.
The area, where the game takes place, is the Cartesian plane(link).
There are N interesting points on the plane. Each interesting point has integer coordinates. Also, all the interesting points form a strictly convex polygon(link).
There is no need to explain the rules of the game, but it's quite important to mention the preparation process:
In order to play the game, the players(Max, K.W. and Carol) must choose three different interesting points
and plant themselves there. Also, they must choose some positive integer S.
Let's assume, that Max's interesting point is A, K.W.'s  B and Carol's  C.
If the area of triangle ABC is greater than or equal to S, then the game will be awesome.
Otherwise, it will be awful.
Now Max is on his way home. His head is filled up with the memories about the wonderful game he used to play with
his friends. But there's the thing, that annoys him: he remembers, that he played the game Q times, but
he can't recall any interesting point chosen by him!
Max exactly remembers, that in the beginning of i'th game K.W. chose point B = B_{i},
Carol chose point C = C_{i} and S was equal to S_{i}.
Also, he remembers, that all the games were awesome.
So, you are to calculate the number of different interesting points,
that could be chosen by Max as A in the beginning of i'th game. This number can be equal to zero(in that case we can conclude, that i'th game wasn't awesome  anyway, just output zero).
Input
The first line of a test case contains two integers N and Q,
denoting the number of the interesting points and the number of times Max played the game.
The next N lines contain two integers X_{i} and Y_{i} each, denoting coordinates of i'th interesting point. The points are 1indexed.
The next Q lines contain three integers B_{i}, C_{i} and S_{i} each. In the beginning of i'th game K.W. chose point B_{i}'th interesting point, Carol chose C_{i}'th interesting point and S was equal to S_{i}.
It's guaranteed, that the interesting points form a strictly convex polygon and are given in clockwise or counterclockwise order.
It's also guaranteed, that there are no three vertices in the input that lie on one line and no point is given twice.
Output
For each round of the game, output a single line containing the number of different interesting points, that could be chosen by Max as A in the beginning of the corresponding game.
Constraints
3 ≤ N ≤ 60000;
1 ≤ Q ≤ 250000;
X, Y ≤ 10^{8}, for each interesting point;
1 ≤ B != C ≤ N, S ≤ 10^{18}, for each game.
Example
Input: 4 2 0 0 4 0 7 3 7 7 1 4 1 1 4 100 Output: 2 0
Explanations
For the first query, A could be equal to both 2 and 3;
For the second query, there are no possible variants for A.
Author:  kostya_by 
Tester:  rustinpiece 
Editorial  http://discuss.codechef.com/problems/WTHINGS 
Tags  binarysearch, cook47, kostya_by, mediumhard, polygons 
Date Added:  30052014 
Time Limit:  2 sec 
Source Limit:  50000 Bytes 
Languages:  C, CPP14, JAVA, PYTH, PYTH 3.6, CS2, PAS fpc, PAS gpc, RUBY, PHP, GO, NODEJS, HASK, SCALA, D, PERL, FORT, WSPC, ADA, CAML, ICK, BF, ASM, CLPS, PRLG, ICON, SCM qobi, PIKE, ST, NICE, LUA, BASH, NEM, LISP sbcl, LISP clisp, SCM guile, JS, ERL, TCL, PERL6, TEXT, PYP3, CLOJ, FS 
Comments
 Please login at the top to post a comment.
SUCCESSFUL SUBMISSIONS
Fetching successful submissions
HELP
If you are still having problems, see a sample solution here. 