Bisectors

We all probably know how to find equation of bisectors in Coordinate Geometry. If the equations of twolinesareaix+biy+ci =0andajx+bjy+cj =0,thentheequationsofthebisectorsofthefour angles they create are given by aix + biy + ci ajx + bjy + cj √=±√ a 2i + b 2i a 2j + b 2j Now one has to be quite intelligent to find out for which angles to choose the ‘+’ (plus) sign and for which angles to choose the ‘-’ (minus) sign. You will have to do similar sort of choosing in this problem. Suppose there is a fixed point (Cx,Cy) and there are n (n ≤ 10000) other points around it. No two points from these n points are collinear with (Cx,Cy). If you connect all these point with (Cx,Cy) you will get a star-topology like image made of n lines. The equations of these n lines are also given and only these equations must be used when finding the equation of bisectors. This n lines create n(n − 1)/2 acute or obtuse angles in total and so they have total n(n − 1)/2 bisectors. You have to find out how many of these bisectors have equations formed using the ‘+’ sign. The image below shows an image where n = 5, Cx = 5 and Cy = 2. This image corresponds to the only sample input. Figure: Five lines above create 5(5 − 1)/2 = 10 angles and these angles has 10 bisectors. Of these 10 bisectors, the equation of only 4 are formed using the ‘+’ sign of the formula aix + biy + ci ajx + bjy + cj √=±√ a 2i + b 2i a 2j + b 2j The input file contains maximum 35 sets of inputs. The description of each set is given below: Input

2/2 First line of each set contains three integers Cx, Cy (−10000 ≤ Cx,Cy ≤ 10000) and n (0 ≤ n ≤ 10000). Each of the next n lines contains two integers xi, yi (−20000 ≤ xi, yi ≤ 20000) and a string of the form aix + biy + ci = 0. Here (xi, yi) is the coordinate of a point around (Cx, Cy) and the string denotes the equation of the line segment formed by connecting (Cx,Cy) and (xi,yi). You can assume that (−100000 ≤ ai,bi ≤ 100000) and (−2000000000 ≤ ci ≤ 2000000000). This equation will actually be used to find the equations of bisectors of the angles that this line creates. Input is terminated by a set where the value of n is zero. Output For each set of input produce one line of output. This line contains an integer number P that denotes of the n(n−1) bisector equations how many are formed using the ‘+’ sign in the bisector equation 2 Sample Input 525 12 7 10x-14y-22=0 1 -4 24x-16y-88=0 4 10 32x+4y-168=0 -1 9 56x+48y-376=0 12 -3 -10x-14y+78=0 10 10 0 Sample Output 4 aix + biy + ci ajx + bjy + cj √=±√ a 2i + b 2i a 2j + b 2j