检查给定点是否在三角形内

给定一个三角形的三个角点，再给定一个点P。编写一个函数检查P是否位于三角形内。
例如，考虑以下程序，该函数应该对P(10，15)返回true，对P’(30，15)返回false。

B(10,30)
                / \
               /   \
              /     \
             /   P   \      P'
            /         \
     A(0,0) ----------- C(20,0) 

解决方案：
设三个角的坐标为(x1，y1)，(x2，y2)和(x3，y3)。给定点P的坐标为(x，y)
1)计算给定三角形的面积，即上图中的三角形ABC的面积。面积A = [x1(y2-y3)+ x2(y3-y1)+ x3(y1-y2)] / 2
2)计算三角形PAB的面积。我们可以为此使用相同的公式。将该区域设为A1。
3)计算三角形PBC的面积。将该区域设为A2。
4)计算三角形PAC的面积。将该区域设为A3。
5)如果P位于三角形内，则A1 + A2 + A3必须等于A。

C++

#include 
using namespace std;
 
/* A utility function to calculate area of triangle formed by (x1, y1),
   (x2, y2) and (x3, y3) */
float area(int x1, int y1, int x2, int y2, int x3, int y3)
{
   return abs((x1*(y2-y3) + x2*(y3-y1)+ x3*(y1-y2))/2.0);
}
 
/* A function to check whether point P(x, y) lies inside the triangle formed
   by A(x1, y1), B(x2, y2) and C(x3, y3) */
bool isInside(int x1, int y1, int x2, int y2, int x3, int y3, int x, int y)
{  
   /* Calculate area of triangle ABC */
   float A = area (x1, y1, x2, y2, x3, y3);
 
   /* Calculate area of triangle PBC */  
   float A1 = area (x, y, x2, y2, x3, y3);
 
   /* Calculate area of triangle PAC */  
   float A2 = area (x1, y1, x, y, x3, y3);
 
   /* Calculate area of triangle PAB */   
   float A3 = area (x1, y1, x2, y2, x, y);
   
   /* Check if sum of A1, A2 and A3 is same as A */
   return (A == A1 + A2 + A3);
}
 
/* Driver program to test above function */
int main()
{
   /* Let us check whether the point P(10, 15) lies inside the triangle
      formed by A(0, 0), B(20, 0) and C(10, 30) */
   if (isInside(0, 0, 20, 0, 10, 30, 10, 15))
     printf ("Inside");
   else
     printf ("Not Inside");
 
   return 0;
}

Java

// JAVA Code for Check whether a given point
// lies inside a triangle or not
import java.util.*;
 
class GFG {
     
    /* A utility function to calculate area of triangle
       formed by (x1, y1) (x2, y2) and (x3, y3) */
    static double area(int x1, int y1, int x2, int y2,
                                        int x3, int y3)
    {
       return Math.abs((x1*(y2-y3) + x2*(y3-y1)+
                                    x3*(y1-y2))/2.0);
    }
      
    /* A function to check whether point P(x, y) lies
       inside the triangle formed by A(x1, y1),
       B(x2, y2) and C(x3, y3) */
    static boolean isInside(int x1, int y1, int x2,
                int y2, int x3, int y3, int x, int y)
    {  
       /* Calculate area of triangle ABC */
        double A = area (x1, y1, x2, y2, x3, y3);
      
       /* Calculate area of triangle PBC */ 
        double A1 = area (x, y, x2, y2, x3, y3);
      
       /* Calculate area of triangle PAC */ 
        double A2 = area (x1, y1, x, y, x3, y3);
      
       /* Calculate area of triangle PAB */  
        double A3 = area (x1, y1, x2, y2, x, y);
        
       /* Check if sum of A1, A2 and A3 is same as A */
        return (A == A1 + A2 + A3);
    }
     
    /* Driver program to test above function */
    public static void main(String[] args)
    {
        /* Let us check whether the point P(10, 15)
           lies inside the triangle formed by
           A(0, 0), B(20, 0) and C(10, 30) */
       if (isInside(0, 0, 20, 0, 10, 30, 10, 15))
           System.out.println("Inside");
       else
           System.out.println("Not Inside");
             
    }
}
 
// This code is contributed by Arnav Kr. Mandal.

Python

# A utility function to calculate area
# of triangle formed by (x1, y1),
# (x2, y2) and (x3, y3)
 
def area(x1, y1, x2, y2, x3, y3):
 
    return abs((x1 * (y2 - y3) + x2 * (y3 - y1)
                + x3 * (y1 - y2)) / 2.0)
 
 
# A function to check whether point P(x, y)
# lies inside the triangle formed by
# A(x1, y1), B(x2, y2) and C(x3, y3)
def isInside(x1, y1, x2, y2, x3, y3, x, y):
 
    # Calculate area of triangle ABC
    A = area (x1, y1, x2, y2, x3, y3)
 
    # Calculate area of triangle PBC
    A1 = area (x, y, x2, y2, x3, y3)
     
    # Calculate area of triangle PAC
    A2 = area (x1, y1, x, y, x3, y3)
     
    # Calculate area of triangle PAB
    A3 = area (x1, y1, x2, y2, x, y)
     
    # Check if sum of A1, A2 and A3
    # is same as A
    if(A == A1 + A2 + A3):
        return True
    else:
        return False
 
# Driver program to test above function
# Let us check whether the point P(10, 15)
# lies inside the triangle formed by
# A(0, 0), B(20, 0) and C(10, 30)
if (isInside(0, 0, 20, 0, 10, 30, 10, 15)):
    print('Inside')
else:
    print('Not Inside')
 
# This code is contributed by Danish Raza

C#

// C# Code to Check whether a given point
// lies inside a triangle or not
using System;
 
class GFG {
 
    /* A utility function to calculate area of triangle
    formed by (x1, y1) (x2, y2) and (x3, y3) */
    static double area(int x1, int y1, int x2,
                       int y2, int x3, int y3)
    {
        return Math.Abs((x1 * (y2 - y3) +
                         x2 * (y3 - y1) +
                         x3 * (y1 - y2)) / 2.0);
    }
 
    /* A function to check whether point P(x, y) lies
    inside the triangle formed by A(x1, y1),
    B(x2, y2) and C(x3, y3) */
    static bool isInside(int x1, int y1, int x2,
                         int y2, int x3, int y3,
                         int x, int y)
    {
        /* Calculate area of triangle ABC */
        double A = area(x1, y1, x2, y2, x3, y3);
 
        /* Calculate area of triangle PBC */
        double A1 = area(x, y, x2, y2, x3, y3);
 
        /* Calculate area of triangle PAC */
        double A2 = area(x1, y1, x, y, x3, y3);
 
        /* Calculate area of triangle PAB */
        double A3 = area(x1, y1, x2, y2, x, y);
 
        /* Check if sum of A1, A2 and A3 is same as A */
        return (A == A1 + A2 + A3);
    }
 
/* Driver program to test above function */
public static void Main()
{
    /* Let us check whether the point P(10, 15)
    lies inside the triangle formed by
    A(0, 0), B(20, 0) and C(10, 30) */
    if (isInside(0, 0, 20, 0, 10, 30, 10, 15))
        Console.WriteLine("Inside");
    else
        Console.WriteLine("Not Inside");
}
}
 
// This code is contributed by vt_m.

Javascript

 
/* A utility function to calculate area of triangle formed by (x1, y1),
(x2, y2) and (x3, y3) */
function area(x1, y1, x2, y2, x3, y3)
{
return Math.abs((x1*(y2-y3) + x2*(y3-y1)+ x3*(y1-y2))/2.0);
}
 
/* A function to check whether point P(x, y) lies inside the triangle formed
by A(x1, y1), B(x2, y2) and C(x3, y3) */
function isInside(x1, y1, x2, y2, x3, y3, x, y)
{
/* Calculate area of triangle ABC */
let A = area (x1, y1, x2, y2, x3, y3);
 
/* Calculate area of triangle PBC */
let A1 = area (x, y, x2, y2, x3, y3);
 
/* Calculate area of triangle PAC */
let A2 = area (x1, y1, x, y, x3, y3);
 
/* Calculate area of triangle PAB */   
let A3 = area (x1, y1, x2, y2, x, y);
     
/* Check if sum of A1, A2 and A3 is same as A */
return (A == A1 + A2 + A3);
}
 
/* Driver program to test above function */
  
/* Let us check whether the point P(10, 15) lies inside the triangle
    formed by A(0, 0), B(20, 0) and C(10, 30) */
if (isInside(0, 0, 20, 0, 10, 30, 10, 15))
    document.write("Inside");
else
    document.write("Not Inside");
 
// This code is contibuted by Mayank Tyagi
 

输出：

Inside 

https://www.youtube.com/watch?v=H9qu9Xptf

-w