Concurrent Lines – Point of concurrency – Examples

Concurrent Lines

When there exists a common point that lies on three or more lines, then these lines are concurrent with each other.

These lines meet each other at the point of concurrency.

In the figure given below

1. Lines AB, CD and PQ have a common point ‘O’ and hence are concurrent.
2. ‘O’ is the point of concurrency.

Difference between concurrent lines and intersecting lines

In intersecting lines, two lines meet each other at a common point known as the point of intersection.

In the case of concurrent lines, three or more lines meet each other at a point known as the point of concurrency

Point of concurrency in a triangle

Incentre

The point of concurrency of all the three angular bisectors of a triangle.

Circumcentre

The point of concurrency of the perpendicular bisectors of the three sides of the triangle.

Centroid

The point of concurrency of all the medians of a triangle.

Orthocentre

The point of concurrency of all the altitudes present in a triangle.

How to check if any three lines are concurrent

First, solve for the point of intersection of two lines and then put the point in the equation of the third line to check if it lies on the third line.

Solved Example

1. The equation of three concurrent lines is given below.

2x + y = 0

3x + 2y + 1 = 0

y = px

Find the value of  ‘p’.

Solution:

First we have to find the point where the first two lines meet each other

2x + y = 0

Multiplying 2 to the given equation: 

⇒ 4x + 2y = 0           ——-   (1) 

3x + 2y + 1 =0       ——-  (2)

Subtracting (2) from (1)

(4x + 2y) – (3x + 2y + 1) = 0

⇒ x – 1 = 0

  ∴ x = 1

Now, we have to substitute the obtained value of x in equation (1)

4x + 2y = 0

4(1) + 2y = 0

⇒ 2y = (-4)

   ∴ y = (-2)

Therefore, if the point (1,-2) is the point of concurrency of these three lines, this point will also satisfy the equation of the third line (y = px)

      y = px

⇒ -2 = p(1)

  ∴ p = (-2)

Hence the value of  ‘p’ is equal to (-2).