#include <cstdio>#include <cstring>#include <cmath>#include <string>#include <iostream>#include <algorithm>#include <queue>#include <set>#include <map>using namespace std;typedef long long ll;#define pii pair<int, int>#define mkpii make_pair<int, int>#define pdi pair<double, int>#define mkpdi make_pair<double, int>#define pli pair<ll, int>#define mkpli make_pair<ll, int>#define rep(i, n) for(int i=0; i<(n); ++i)#define for1(i,a,n) for(int i=(a);i<=(n);++i)#define for2(i,a,n) for(int i=(a);i<(n);++i)#define for3(i,a,n) for(int i=(a);i>=(n);--i)#define for4(i,a,n) for(int i=(a);i>(n);--i)#define CC(i,a) memset(i,a,sizeof(i))#define read(a) a=getint()#define print(a) printf("%d", a)#define dbg(x) cout << (#x) << " = " << (x) << endl#define error(x) (!(x)?puts("error"):0)#define printarr2(a, b, c) for1(_, 1, b) { for1(__, 1, c) cout << a[_][__]; cout << endl; }#define printarr1(a, b) for1(_, 1, b) cout << a[_] << ‘\t‘; cout << endlinline const int getint() { int r=0, k=1; char c=getchar(); for(; c<‘0‘||c>‘9‘; c=getchar()) if(c==‘-‘) k=-1; for(; c>=‘0‘&&c<=‘9‘; c=getchar()) r=r*10+c-‘0‘; return k*r; }inline const int max(const int &a, const int &b) { return a>b?a:b; }inline const int min(const int &a, const int &b) { return a<b?a:b; }struct Point { int x, y; }a[4];int area(Point &a, Point &b, Point &c) {	static int x1, x2, y1, y2;	x1=a.x-c.x; y1=a.y-c.y;	x2=b.x-c.x; y2=b.y-c.y;	return abs(x1*y2-x2*y1);}int gcd(int a, int b) { return b?gcd(b, a%b):a; }int onLine(Point &a, Point &b) {	static int x, y;	x=abs(a.x-b.x);	y=abs(a.y-b.y);	return gcd(x, y);}int main() {	while(1) {		int sum=0;		rep(i, 3) read(a[i].x), read(a[i].y), sum|=a[i].x|a[i].y;		if(!sum) return 0;		int on=onLine(a[0], a[1])+onLine(a[1], a[2])+onLine(a[2], a[0]);		printf("%d\n", (area(a[0], a[1], a[2])-on)/2+1);	}	return 0;}

A lattice point is an ordered pair (x, y) where x and y are both integers. Given the coordinates of the vertices of a triangle (which happen to be lattice points), you are to count the number of lattice points which lie completely inside of the triangle (points on the edges or vertices of the triangle do not count).

The input test file will contain multiple test cases. Each input test case consists of six integers x₁, y₁, x₂, y₂, x₃, and y₃, where (x₁, y₁), (x₂, y₂), and (x₃, y₃) are the coordinates of vertices of the triangle. All triangles in the input will be non-degenerate (will have positive area), and −15000 ≤ x₁, y₁, x₂, y₂, x₃, y₃ ≤ 15000. The end-of-file is marked by a test case with x₁ =  y₁ = x₂ = y₂ = x₃ = y₃ = 0 and should not be processed.

For each input case, the program should print the number of internal lattice points on a single line.