# UVA 11800 – Determine the Shape(计算几何)

#include <cstdio>
#include <cstring>

#include<cstdio>
#include<cmath>
#include<algorithm>
using namespace std;

const int MAXN = 4;

struct Point {
int x, y;
Point() {}
Point(double x, double y) {
this->x = x;
this->y = y;
}
scanf("%d%d", &x, &y);
}
} list[MAXN], p[MAXN];

typedef Point Vector;

Vector operator + (Vector A, Vector B) {
return Vector(A.x + B.x, A.y + B.y);
}

Vector operator – (Vector A, Vector B) {
return Vector(A.x – B.x, A.y – B.y);
}

Vector operator * (Vector A, double p) {
return Vector(A.x * p, A.y * p);
}

Vector operator / (Vector A, double p) {
return Vector(A.x / p, A.y / p);
}

int Dot(Vector A, Vector B) {return A.x * B.x + A.y * B.y;} //点积
int Cross(Vector A, Vector B) {return A.x * B.y – A.y * B.x;} //叉积

int stack[MAXN], top;

int xmult(Point p0, Point p1, Point p2) {
return (p1.x – p0.x) * (p2.y – p0.y) – (p1.y – p0.y) * (p2.x – p0.x);
}

double dist(Point p1,Point p2) {
return sqrt((double)(p2.x-p1.x)*(p2.x-p1.x)+(p2.y-p1.y)*(p2.y-p1.y));
}

int dist2(Point p1, Point p2) {
return (p2.x – p1.x) * (p2.x – p1.x) + (p2.y – p1.y) * (p2.y – p1.y);
}

//极角排序函数 ， 角度相同则距离小的在前面
bool cmp(Point p1,Point p2) {
int tmp= xmult(list[0], p1, p2);
if(tmp > 0) return true;
else if(tmp == 0 && dist(list[0], p1) < dist(list[0], p2)) return true;
else return false;
}

bool LineParallel(Vector v, Vector w) {
return Cross(v, w) == 0;
}

bool LineVertical(Vector v, Vector w) {
return Dot(v, w) == 0;
}

//输入并把最左下方的点放在list[0]并且进行极角排序
void init(int n) {
int i, k;
Point p0;
scanf("%d%d", &list[0].x, &list[0].y);
p0.x = list[0].x;
p0.y = list[0].y;
k = 0;
for(i = 1; i < n; i++) {
scanf("%d%d", &list[i].x, &list[i].y);
if((p0.y > list[i].y) || ((p0.y == list[i].y) && (p0.x > list[i].x))) {
p0.x = list[i].x;
p0.y = list[i].y;
k = i;
}
}
list[k] = list[0];
list[0] = p0;
sort(list + 1, list + n, cmp);
}

void graham(int n) {
int i;
if(n == 1) {top = 0; stack[0] = 0;}
if(n == 2) {
top = 1;
stack[0] = 0;
stack[1] = 1;
}
if(n > 2) {
for(i = 0; i <= 1; i++) stack[i] = i;
top = 1;
for(i = 2; i < n; i++) {
while(top > 0 && xmult(list[stack[top – 1]], list[stack[top]], list[i]) <= 0) top–;
top++;
stack[top]=i;
}
}
for (int i = 0; i <= top; i++)
p[i] = list[stack[i]];
}

int t;

void solve() {
if (LineParallel(p[1] – p[0], p[2] – p[3]) && LineParallel(p[2] – p[1], p[3] – p[0])) {
if (LineVertical(p[1] – p[0], p[2] – p[1]) && LineVertical(p[2] – p[1], p[3] – p[2]) && LineVertical(p[0] – p[3], p[3] – p[2])) {
if (dist2(p[1], p[0]) == dist2(p[2], p[1])) printf("Square
");
else printf("Rectangle
");
} else {
if (dist2(p[1], p[0]) == dist2(p[2], p[1])) printf("Rhombus
");
else printf("Parallelogram
");
}
} else if (LineParallel(p[1] – p[0], p[2] – p[3]) || LineParallel(p[2] – p[1], p[3] – p[0])) printf("Trapezium
");
");
}

int main() {
int cas = 0;
scanf("%d", &t);
while (t–) {
init(4);
graham(4);
printf("Case %d: ", ++cas);
if (top < 3) {
");
continue;
}
for (int i = 0; i < 4; i++)
p[i] = list[stack[i]];
solve();
}
return 0;
}

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