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最新推荐文章于 2019-08-25 00:45:00 发布

原创最新推荐文章于 2019-08-25 00:45:00 发布 · 1k 阅读

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#计算几何学 #ACM题解报告

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本文介绍如何利用三角形的三条边长计算其外接圆的半径及外心坐标，并提供了完整的C++代码实现。文章还涵盖了半周长、面积等基本几何概念及其计算方法。

三角形外心及外接圆半径

三角形三边为 a、b、c
半周长 p=(a+b+c)/2
三角形面积 S=√[p(p-a)(p-b)(p-c)] （海伦公式）
内切圆半径 r = S/p
=√[(p-a)(p-b)(p-c)/p]
= ½√[(-a+b+c)(a-b+c)(a+b-c)/(a+b+c)]
外接圆半径 R= abc/(4S)
= ¼ abc/√[p(p-a)(p-b)(p-c)]
= abc/√[(a+b+c)(-a+b+c)(a-b+c)(a+b-c)]
R、r、S 关系
rR = S/p * abc/(4S) = abc/[2(a+b+c)]

#include <iostream>
#include <math.h>
#include <stdio.h>

#define pi acos(-1.0)

struct point{ double x, y; };
struct line{ point a, b; };

double xmult(point p1, point p2, point p0)
{
	return (p1.x - p0.x)*(p2.y - p0.y) - (p2.x - p0.x)*(p1.y - p0.y);
}

char sign(double a)
{
	return a > 0 ? '-' : '+';
}
char sign2(double a)
{
	return a > 0 ? '+' : '-';
}

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

point intersection(line u, line v){
	point ret = u.a;
	double t = ((u.a.x - v.a.x)*(v.a.y - v.b.y) - (u.a.y - v.a.y)*(v.a.x - v.b.x))
		/ ((u.a.x - u.b.x)*(v.a.y - v.b.y) - (u.a.y - u.b.y)*(v.a.x - v.b.x));
	ret.x += (u.b.x - u.a.x)*t;
	ret.y += (u.b.y - u.a.y)*t;
	return ret;
}

//外心
point circumcenter(point a, point b, point c){
	line u, v;
	u.a.x = (a.x + b.x) / 2;
	u.a.y = (a.y + b.y) / 2;
	u.b.x = u.a.x - a.y + b.y;
	u.b.y = u.a.y + a.x - b.x;
	v.a.x = (a.x + c.x) / 2;
	v.a.y = (a.y + c.y) / 2;
	v.b.x = v.a.x - a.y + c.y;
	v.b.y = v.a.y + a.x - c.x;
	return intersection(u, v);
}

double area_triangle(point p1, point p2, point p3)
{
	return fabs(xmult(p1, p2, p3)) / 2;
}

int main()
{
	point a, b, c;
	while (std::cin >> a.x >> a.y >> b.x >> b.y >> c.x >> c.y)
	{
		point center = circumcenter(a, b, c);
		double p = distance(a, b) * distance(a, c) * distance(b, c);
		double r = 1.0 * p / (4.0*area_triangle(a, b, c));
		printf("(x %c %.3f)^2 + (y %c %.3f)^2 = %.3f^2\n", sign(center.x), fabs(center.x), sign(center.y), fabs(center.y),r);
		printf("x^2 + y^2 %c %.3fx %c %.3fy %c %.3f = 0\n", sign(center.x), 2 * fabs(center.x), sign(center.y), 2 * fabs(center.y), sign2(center.x * center.x + center.y * center.y - r * r), fabs(center.x * center.x + center.y * center.y - r * r));
		puts("");
	}
}