Check if three straight lines are concurrent or not

Difficulty Level : Easy
Last Updated : 25 Jun, 2018

Given three lines equation,
a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0
a₃x + b₃y + c₃ = 0

The task is to check whether the given three lines are concurrent or not. Three straight lines are said to be concurrent if they pass through a point i.e., they meet at a point.

Examples:

Input : a₁ = 2, b₁ = -3, c₁ = 5
        a₂ = 3, b₂ = 4, c₂ = -7
        a₃ = 9, b₃ = -5, c₃ = 8
Output : Yes

Input : a₁ = 2, b₁ = -3, c₁ = 5
        a₂ = 3, b₂ = 4, c₂ = -7
        a₃ = 9, b₃ = -5, c₃ = 4
Output : No

Let
a₁x + b₁y + c₁ = 0 ………. (1)
a₂x + b₂y + c₂ = 0 ………. (2)
a₃x + b₃y + c₃ = 0 ………. (3)

Suppose the eqn (i) and (ii) intersets at (x₁, y₁). Then (x₁, y₁) will satisfy bothe equations.
Therefore, solving (i) and (ii) using method of cross-multiplication, we get,
(x₁/b₁c₂ – b₂c₁) = (y₁/c₁a₂ – c₂a₁) = (1/a₁b₂ – a₂b₁)

Therefore,
x₁ = (b₁c₂ – b₂c₁/a₁b₂ – a₂b₁) and
y₁ = (c₁a₂ – c₂a₁/a₁b₂ – a₂b₁), a₁b₂ – a₂b₁ != 0

Therefore, the required coodinates of the point of intersection of the lines (i) and (ii) are
(b₁c₂ – b₂c₁/a₁b₂ – a₂b₁, c₁a₂ – c₂a₁/a₁b₂ – a₂b₁)

For, three of line to be concurrent, (x₁, y₁) must satisfy the equation (iii) as well.
So,
a₃x + b₃y + c₃ = 0
=> a₃(b₁c₂ – b₂c₁/a₁b₂ – a₂b₁) + b₃(c₁a₂ – c₂a₁/a₁b₂ – a₂b₁) + c₃ = 0
=> a₃(b₁c₂ – b₂c₁) + b₃(c₁a₂ – c₂a₁) + c₃(a₁b₂ – a₂b₁) = 0

So, we only need to check if above condition satisfy or not.

Below is the implemenatation of this approach:

C++

// CPP Program to check if three straight  
// line are concurrent or not 
#include <bits/stdc++.h> 
using namespace std; 
  
// Return true if three line are concurrent, 
// else false. 
bool checkConcurrent(int a1, int b1, int c1,  
                      int a2, int b2, int c2,  
                      int a3, int b3, int c3) 
{ 
    return (a3 * (b1 * c2 - b2 * c1) +  
            b3 * (c1 * a2 - c2 * a1) +   
            c3 * (a1 * b2 - a2 * b1) == 0); 
} 
  
// Driven Program 
int main() 
{ 
    int a1 = 2, b1 = -3, c1 = 5; 
    int a2 = 3, b2 = 4, c2 = -7; 
    int a3 = 9, b3 = -5, c3 = 8; 
  
    (checkConcurrent(a1, b1, c1, a2, b2, c2,  
     a3, b3, c3) ? (cout << "Yes") : (cout << "No")); 
    return 0; 
} 

Java

// Java Program to check if three straight  
// line are concurrent or no 
import java.io.*; 
  
class GFG { 
  
    // Return true if three line are concurrent, 
    // else false. 
    static boolean checkConcurrent(int a1, int b1, 
                   int c1, int a2, int b2, int c2,  
                           int a3, int b3, int c3) 
    { 
        return (a3 * (b1 * c2 - b2 * c1) +  
                b3 * (c1 * a2 - c2 * a1) +  
                c3 * (a1 * b2 - a2 * b1) == 0); 
    } 
      
    // Driven Program 
    public static void main (String[] args)  
    { 
        int a1 = 2, b1 = -3, c1 = 5; 
        int a2 = 3, b2 = 4, c2 = -7; 
        int a3 = 9, b3 = -5, c3 = 8; 
      
        if(checkConcurrent(a1, b1, c1, a2, b2, 
                               c2, a3, b3, c3))  
            System.out.println( "Yes"); 
        else
            System.out.println( "No"); 
    } 
} 
  
// This code is contributed by anuj_67. 

Python 3

# Python3 Program to check if three straight  
# line are concurrent or not 
  
# Return true if three line are concurrent, 
# else false. 
def checkConcurrent(a1, b1, c1, a2, b2, c2, 
                                 a3, b3, c3): 
  
    return (a3 * (b1 * c2 - b2 * c1) +
            b3 * (c1 * a2 - c2 * a1) +
            c3 * (a1 * b2 - a2 * b1) == 0) 
  
  
# Driven Program 
a1 = 2
b1 = -3
c1 = 5
a2 = 3
b2 = 4
c2 = -7
a3 = 9
b3 = -5
c3 = 8
  
if(checkConcurrent(a1, b1, c1, a2, b2, c2, 
                               a3, b3, c3)): 
    print("Yes") 
else: 
    print("No") 
  
# This code is contributed by Smitha 

C#

// C# Program to check if three straight  
// line are concurrent or no 
using System; 
  
class GFG { 
  
    // Return true if three line are concurrent, 
    // else false. 
    static bool checkConcurrent(int a1, int b1, 
                int c1, int a2, int b2, int c2,  
                        int a3, int b3, int c3) 
    { 
        return (a3 * (b1 * c2 - b2 * c1) +  
                b3 * (c1 * a2 - c2 * a1) +  
                c3 * (a1 * b2 - a2 * b1) == 0); 
    } 
      
    // Driven Program 
    public static void Main ()  
    { 
        int a1 = 2, b1 = -3, c1 = 5; 
        int a2 = 3, b2 = 4, c2 = -7; 
        int a3 = 9, b3 = -5, c3 = 8; 
      
        if(checkConcurrent(a1, b1, c1, a2, b2, 
                            c2, a3, b3, c3))  
            Console.WriteLine( "Yes"); 
        else
            Console.WriteLine( "No"); 
    } 
} 
  
// This code is contributed by anuj_67. 

PHP

<?php 
// PHP Program to check if three straight  
// line are concurrent or not 
  
// Return true if three line are  
// concurrent, else false. 
function checkConcurrent($a1, $b1, $c1,  
                         $a2, $b2, $c2,  
                         $a3, $b3, $c3) 
{ 
    return ($a3 * ($b1 * $c2 - $b2 * $c1) +  
            $b3 * ($c1 * $a2 - $c2 * $a1) +  
            $c3 * ($a1 * $b2 - $a2 * $b1) == 0); 
}    
  
    // Driver Code 
    $a1 = 2; $b1 = -3; $c1 = 5; 
    $a2 = 3; $b2 = 4; $c2 = -7; 
    $a3 = 9; $b3 = -5; $c3 = 8; 
  
    if(checkConcurrent($a1, $b1, $c1, $a2, $b2, 
                           $c2, $a3, $b3, $c3)) 
        echo "Yes"; 
    else
        echo "No"; 
  
// This code is contributed by anuj_67. 
?> 

Output:

Yes

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