What is Concurrent lines in Geometry – Definition, Conditions & Examples

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The study of lines is one of the important topics in geometry.  Two or more lines in a plane can be parallel lines or intersecting lines. When two or more lines pass through the same point, they are called concurrent lines.

Let’s understand what is concurrent lines in geometry, and conditions when the lines are concurrent with examples.

What is Concurrent lines in Geometry?

Concurrent lines are defined as a set of lines that intersect at a common point. Three or more lines are said to be concurrent if they intersect at a point. Only lines can be concurrent, rays and line segments can not be concurrent since they do not necessarily meet at a point all the time. There can be more than two lines that pass through a point. 

Some of the few examples are the diameters of a circle are concurrent at the center of the circle. In quadrilaterals, the line segments joining the midpoints of opposite sides, and the diagonals are concurrent.

Difference Between Concurrent Lines and Intersecting Lines

As discussed above, if any three lines or line segments or rays are having a single intersection point, they are said to be in concurrency. While, in the case of intersecting lines, there are only two lines or line segments or rays that intersect with each other. 

The following are the difference between concurrent lines and intersecting lines

Concurrent Lines	Intersecting LInes
Three or more lines pass through a single point.	Only two lines intersect each other.
The single point at which these lines intersect each other is called a point of concurrency.	The point where two lines intersect is called the point of intersection.
Example:	Example:

Concurrent Lines in Triangles

In a triangle, the concurrent lines are:

• Altitudes: The three altitudes of a triangle from all three vertices intersect each other at a common point. This point where the altitudes intersect is called the orthocenter.
• Medians: The three medians of a triangle that divides the opposite side into equal parts and intersects at a single point, known as the centroid.
• Angle Bisectors: Angle bisectors are the rays that bisect the angle from each vertex and meet at a single point. The point is called here as incenter.
• Perpendicular Bisectors: The perpendicular bisectors are the lines that intersect the opposite sides at $90^{\circ}$ angles and pass through a single point. This point is called the circumcenter.

How to Find If Lines are Concurrent?

There are two methods to check whether given lines are concurrent or not. 

Method 1

Let us consider three lines,

Line 1 = $a_1x  + b_1y + c_1z  = 0$ and 

Line 2 = $a_2x  + b_2y + c_2z  = 0$ and 

Line 3 = $a_3x  + b_3y + c_3z  = 0$

To check if the above three lines are concurrent, the following condition as a determinant should be evaluated to $0$.

Examples

Ex 1: Check whether the lines represented by the following equations are concurrent or not.

$4x – 2y – 4 = 0$ —– (1)

$x – y + 2 = 0$         —– (2)

$2x + 3y – 26 = 0$    —– (3) 

Here, $a_1 = 4$, $b_1 = -2$, and $c_1 = -4$

$a_2 = 1$, $b_2 = -1$, and $c_2 = 2$

$a_3 = 2$, $b_3 = 3$, and $c_3 = -26$

Substituting the above values in the formula we get

$= 4 \times \left(-1 \times \left(-26 \right) – 2 \times 3 \right) – (-2) \times (1 \times (-26) – 2 \times 2) – 4 \times (1 \times 3 – (-1) \times 2)$

$= 4 \times \left(26 – 6 \right) + 2 \times (-26 – 4) – 4 \times (3 + 2)$

$= 4 \times (20) + 2 \times (-30) – 4 \times 5$

$= 80 – 60 – 20 = 0$

Therefore, the given lines are concurrent.

Method 2

To check if three lines are concurrent, we first find the point of intersection of two lines and then check to see if the third line passes through the intersection point. This will ensure that all three lines are concurrent. Let us understand this better with an example. The equations of any three lines are as follows. 

Examples

Ex 1: Check whether the lines represented by the following equations are concurrent or not.

$4x – 2y – 4 = 0$ —– (1)

$x – y + 2 = 0$         —– (2)

$2x + 3y – 26 = 0$    —– (3) 

Step 1: To find the point of intersection of line 1 and line 2, solve the equations (1) and (2) by substitution method. 

Substituting the value of $y$ from equation (2) in equation (1) we get, 

$4x – 2 (x + 2) – 4 = 0$

$4x – 2x – 4 – 4 = 0$ 

$2x – 8 = 0$

$x = \frac{8}{2}$ 

$x = 4$ 

Substituting the value of $x = 4$ in equation (2), we get the value of $y$. 

$y = x + 2$ 

$y = 4 + 2$ 

$y = 6$

Therefore, line 1 and line 2 intersect at a point $(4,6)$.

Step 2:  Substitute the point of intersection of the first two lines in the equation of the third line. 

Equation of the third line is $2x + 3y = 26$    —– (3) 

Substituting the values of $(4,6)$ in equation (3), we get, 

$2 \times 4 + 3 \times 6 = 26$

$8 + 18 = 26$

$26 = 26$ 

Therefore, the point of intersection goes right with the third line equation, which means the three lines intersect each other and are concurrent lines.

Practice Problems

1. What are concurrent lines?
2. What is the difference between intersecting lines and concurrent lines?
3. What is the condition for three lines to be concurrent?
4. Fill in the blanks
   • The point through which all the concurrent lines pass is called __________.
   • The point of concurrency of three medians of a triangle is called _________.
   • The point of concurrency of three altitudes of a triangle is called _________.
   • The point of concurrency of three angle bisectors of a triangle is called _________.
   • The point of concurrency of three perpendicular bisectors of a triangle is called _________.

FAQs

Conclusion

When three or more than three lines pass through the same point, the lines are called concurrent lines, and the point of intersection of these lines is called the point of concurrency. Concurrent lines are an important field of study in triangles.

Recommended Reading

• What is Half Line in Geometry – Definition, Properties & Examples
• What is a Perpendicular Line in Geometry – Definition, Properties & Examples
• Difference Between Axiom, Postulate and Theorem
• Lines in Geometry(Definition, Types & Examples)
• What Are 2D Shapes – Names, Definitions & Properties
• 3D Shapes – Definition, Properties & Types