Sha 11178 (simple exercise) Morley's Theorem

Question:

Morley's theorem: In any triangle, the three equal points of each angle form a positive triangle.

However, this has little to do with the question. The question is the coordinates of the three vertices of the positive triangle, with 6 decimal places retained.

Analysis:

Because of the symmetry, EF is the same for finding the D point.

A straight line is expressed in the form of a vertex and a vector. If the angle between the vectors BA and BC is A1, a straight line BD can be obtained by rotating BC counterclockwise A1/3, and a straight line CD can also be obtained, finally, calculate the intersection.

 

  1 //#define LOCAL  2 #include <cstdio>  3 #include <cstring>  4 #include <algorithm>  5 #include <cmath>  6 using namespace std;  7   8 struct Point  9 { 10     double x, y; 11     Point(double x=0, double y=0) :x(x),y(y) {} 12 }; 13 typedef Point Vector; 14 const double EPS = 1e-10; 15  16 Vector operator + (Vector A, Vector B)    { return Vector(A.x + B.x, A.y + B.y); } 17  18 Vector operator - (Vector A, Vector B)    { return Vector(A.x - B.x, A.y - B.y); } 19  20 Vector operator * (Vector A, double p)    { return Vector(A.x*p, A.y*p); } 21  22 Vector operator / (Vector A, double p)    { return Vector(A.x/p, A.y/p); } 23  24 bool operator < (const Point& a, const Point& b) 25 { return a.x < b.x || (a.x == b.x && a.y < b.y); } 26  27 int dcmp(double x) 28 { if(fabs(x) < EPS) return 0; else x < 0 ? -1 : 1; } 29  30 bool operator == (const Point& a, const Point& b) 31 { return dcmp(a.x-b.x) == 0 && dcmp(a.y-b.y) == 0; } 32  33 double Dot(Vector A, Vector B) 34 { return A.x*B.x + A.y*B.y; } 35  36 double Length(Vector A)    { return sqrt(Dot(A, A)); } 37  38 double Angle(Vector A, Vector B) 39 { return acos(Dot(A, B) / Length(A) / Length(B)); } 40  41 double Cross(Vector A, Vector B) 42 { return A.x*B.y - A.y*B.x; } 43  44 double Area2(Point A, Point B, Point C) 45 { return Cross(B-A, C-A); } 46  47 Vector VRotate(Vector A, double rad) 48 { 49     return Vector(A.x*cos(rad) - A.y*sin(rad), A.x*sin(rad) + A.y*cos(rad)); 50 } 51  52 Point GetLineIntersection(Point P, Vector v, Point Q, Vector w) 53 { 54     Vector u = P - Q; 55     double t = Cross(w, u) / Cross(v, w); 56     return P + v*t; 57 } 58  59 Point read_point(void) 60 { 61     double x, y; 62     scanf("%lf%lf", &x, &y); 63     return Point(x, y); 64 } 65  66 Point GetD(Point A, Point B, Point C) 67 { 68     Vector v1 = C - B; 69     double a1 = Angle(A-B, v1); 70     v1 = VRotate(v1, a1/3); 71  72     Vector v2 = B - C; 73     double a2 = Angle(A-C, v2); 74     v2 = VRotate(v2, -a2/3); 75  76     return GetLineIntersection(B, v1, C, v2); 77 } 78  79 int main(void) 80 { 81     #ifdef    LOCAL 82         freopen("11178in.txt", "r", stdin); 83     #endif 84      85     int T; 86     scanf("%d", &T); 87     while(T--) 88     { 89         Point A, B, C, D, E, F; 90         A = read_point(); 91         B = read_point(); 92         C = read_point(); 93         D = GetD(A, B, C); 94         E = GetD(B, C, A); 95         F = GetD(C, A, B); 96         printf("%.6lf %.6lf %.6lf %.6lf %.6lf %.6lf\n", D.x, D.y, E.x, E.y, F.x, F.y); 97     } 98  99     return 0;100 }

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Sha 11178 (simple exercise) Morley's Theorem