2019 ICPC Asia Nanjing Regional K. Triangle（计算几何+二分）

 把板子都抄上了所以代码特别长，其实只用到了几个函数，先判断给定的点p是不是在顶点上，如果是的是的话直接输出对边的中点，如果正好在某条边的中点上的话，同理。

另外情况需要二分，如果点在ab上且在靠近a的地方，那么另一个点在bc上，否则在ac上，其余边上同理，然后二分找就可以了，叉积算三角形面积。

// #pragma GCC optimize(2)
#include <algorithm>
#include <iostream>
#include <sstream>
#include <cstring>
#include <cstdio>
#include <cctype>
#include <bitset>
#include <string>
#include <vector>
#include <queue>
#include <cmath>
#include <stack>
#include <set>
#include <map>
#define IO                       \
	ios::sync_with_stdio(false); \
	// cout.tie(0);
#define lson(x) (x << 1)
#define rson(x) (x << 1 | 1)
using namespace std;
// int dis[8][2] = {0, 1, 1, 0, 0, -1, -1, 0, 1, -1, 1, 1, -1, 1, -1, -1};
typedef unsigned long long ULL;
typedef long long LL;
typedef pair<int, int> P;
const int maxn = 1e6 + 10;
const int maxm = 1e7 + 10;
const LL INF = 0x3f3f3f3f3f3f3f3f;
const int inf = 0x3f3f3f3f;
const LL mod = 998244353;
const double eps = 1e-8;
const double pi = acos(-1);
// int dis[4][2] = {1, 0, 0, -1, 0, 1, -1, 0};
// int m[13] = {0, 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31};
int cmp(double x)
{
	if (fabs(x) < eps)
		return 0;
	if (x > 0)
		return 1;
	return -1;
}
inline double sqr(double x)
{
	return x * x;
}
struct point
{
	double x, y;
	point() {}
	point(double a, double b) : x(a), y(b) {}
	friend point operator+(const point &a, const point &b)
	{
		return point(a.x + b.x, a.y + b.y);
	}
	friend point operator-(const point &a, const point &b)
	{
		return point(a.x - b.x, a.y - b.y);
	}
	friend bool operator==(const point &a, const point &b)
	{
		return cmp(a.x - b.x) == 0 && cmp(a.y - b.y) == 0;
	}
	friend point operator*(const point &a, const double &b)
	{
		return point(a.x * b, a.y * b);
	}
	friend point operator*(const double &a, const point &b)
	{
		return point(a * b.x, a * b.y);
	}
	friend point operator/(const point &a, const double &b)
	{
		return point(a.x / b, a.y / b);
	}
	double norm()
	{
		return sqrt(sqr(x) + sqr(y));
	}
} a, b, c, p;
double det(const point &a, const point &b)
{
	return a.x * b.y - a.y * b.x;
}
double dot(const point &a, const point &b)
{
	return a.x * b.x + a.y * b.y;
}
double dist(const point &a, const point &b)
{
	return (a - b).norm();
}
point rotate_point(const point &p, double A)
{
	double tx = p.x, ty = p.y;
	return point(tx * cos(A) - ty * sin(A), tx * sin(A) + ty * cos(A));
}
struct line
{
	point a, b;
	line() {}
	line(point x, point y) : a(x), b(y) {}
};
line point_make_line(const point a, const point b)
{
	return line(a, b);
}
double dis_point_segment(const point p, const point s, point t)
{
	if (cmp(dot(p - s, t - s)) < 0)
		return (p - s).norm();
	if (cmp(dot(p - t, s - t)) < 0)
		return (p - t).norm();
	return fabs(det(s - p, t - p) / dist(s, t));
}
void PointProjline(const point p, const point s, const point t, point &cp)
{
	double r = dot((t - s), (p - s)) / dot(t - s, t - s);
	cp = s + r * (t - s);
}
bool PointOnSegment(point p, point s, point t)
{
	return cmp(det(p - s, t - s)) == 0 && cmp(dot(p - s, p - t)) <= 0;
}
bool parallel(line a, line b)
{
	return !cmp(det(a.a - a.b, b.a - b.b));
}
bool line_make_point(line a, line b, point &res)
{
	if (parallel(a, b))
		return false;
	double s1 = det(a.a - b.a, b.b - b.a);
	double s2 = det(a.b - b.a, b.b - b.a);
	res = (s1 * a.b - s2 * a.a) / (s1 - s2);
	return true;
}
line move_d(line a, const double &len)
{
	point d = a.b - a.a;
	d = d / d.norm();
	d = rotate_point(d, pi / 2);
	return line(a.a + d * len, a.b + d * len);
}
double s;

point BinSear(point p, point L, point R)
{
	point pp = R;
	point mid;
	for (int i = 0; i < 100; i++)
	{
		mid = (L + R) / 2.0;
		double area = fabs(det(pp - mid, pp - p));
		if (cmp(area - s) > 0)
			L = mid;
		else if (cmp(area - s) < 0)
			R = mid;
		else
			break;
	}
	return mid;
}
int main()
{
#ifdef WXY
	freopen("in.txt", "r", stdin);
	//  freopen("out.txt", "w", stdout);
#endif
	// IO;
	int T;
	scanf("%d", &T);
	while (T--)
	{
		scanf("%lf %lf %lf %lf %lf %lf %lf %lf", &a.x, &a.y, &b.x, &b.y, &c.x, &c.y, &p.x, &p.y);
		s = abs(det(a - c, a - b) * 0.5);
		if (p == a)
		{
			point t = (b + c) / 2;
			printf("%.8lf %.8lf\n", t.x, t.y);
			continue;
		}
		if (p == b)
		{
			point t = (a + c) / 2;
			printf("%.8lf %.8lf\n", t.x, t.y);
			continue;
		}
		if (p == c)
		{
			point t = (b + a) / 2;
			printf("%.8lf %.8lf\n", t.x, t.y);
			continue;
		}
		if (PointOnSegment(p, a, b))
		{
			point t = (a + b) / 2;
			if (p == t)
			{
				printf("%.8lf %.8lf\n", c.x, c.y);
				continue;
			}
			if (PointOnSegment(p, a, t))
			{
				point ans = BinSear(p, c, b);
				printf("%.8lf %.8lf\n", ans.x, ans.y);
			}
			else
			{
				point ans = BinSear(p, c, a);
				printf("%.8lf %.8lf\n", ans.x, ans.y);
			}
			continue;
		}
		else if (PointOnSegment(p, b, c))
		{
			point t = (b + c) / 2;
			if (p == t)
			{
				printf("%.8lf %.8lf\n", a.x, a.y);
				continue;
			}
			if (PointOnSegment(p, b, t))
			{
				point ans = BinSear(p, a, c);
				printf("%.8lf %.8lf\n", ans.x, ans.y);
			}
			else
			{
				point ans = BinSear(p, a, b);
				printf("%.8lf %.8lf\n", ans.x, ans.y);
			}
			continue;
		}
		else if (PointOnSegment(p, a, c))
		{
			point t = (a + c) / 2;
			if (p == t)
			{
				printf("%.8lf %.8lf\n", b.x, b.y);
				continue;
			}
			if (PointOnSegment(p, a, t))
			{
				point ans = BinSear(p, b, c);
				printf("%.8lf %.8lf\n", ans.x, ans.y);
			}
			else
			{
				point ans = BinSear(p, b, a);
				printf("%.8lf %.8lf\n", ans.x, ans.y);
			}
			continue;
		}
		else
		{
			printf("-1\n");
		}
	}
	return 0;
}