Check if three straight lines are concurrent or not

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) intersects at (x₁, y₁). Then (x₁, y₁) will satisfy both the 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 coordinates 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 implementation 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.
?>

Javascript

<script>

// Javascript 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);
}

// Driven Program
a1 = 2, b1 = -3, c1 = 5;
a2 = 3, b2 = 4, c2 = -7;
a3 = 9, b3 = -5, c3 = 8;
(checkConcurrent(a1, b1, c1, a2, b2, c2,
 a3, b3, c3) ? (document.write( "Yes")) : (document.write( "No")));

</script>

Output:

Yes

Time Complexity: O(1)

Auxiliary Space: O(1)