Concurrency of Lines

Introduction:

The term “concurrency of lines” refers to lines in a plane that intersect at the same place. Whenever two lines that are not parallel come together, they form a point of intersection. When a third line crosses the intersection point formed by the first two lines, the intersection is said to be a concurrent line intersection. 

The ‘Point of Concurrency’ is the intersection of all of these lines. As an example, three altitudes drawn on a triangle connect at the ‘Orthocenter’, which is a point in the middle of the triangle. Due to their endless length and eventual convergence, only non parallel lines have a point of concurrence.

How to calculate the concurrency of lines?

There are two ways to determine whether or not three lines are concurrent. Here is how:

1st method:

There are three lines:

• Line 1 = a1x + b1y + c1z = 0 and 
• Line 2 = a2x + b2y + c2z  = 0 and 
• Line 3 = a3x + b3y + c3z = 0.

To determine if the three lines above are concurrent, the following determinant shown below should be evaluated to a value of 0.

a1   b1    c1

a2    b2    c2 =0

a3   b3    c3

You can also use another method to see if the lines intersect.

2nd method:

Once two lines are intersected, the third line must pass through this point to determine if the three lines are congruent. All three lines will be running simultaneously as a result resulting in the concurrency of lines. This will be easier to understand if we look at an example. For any three lines, you have the following equations to solve for each variable. 

• 4x – 2y – 4 = 0 —– (1)
• y = x + 2         —– (2)
• 2x + 3y = 26    —– (3)

1st step: Solve equations (1) and (2) by substitution to find the point of intersection of lines 1 and 2.

• Equation (1) is obtained by substituting the value of ‘y’ from equation (2).

y = x + 2 —– (2)

y = 4 + 2 

y = 6

• Because of this, lines 1 and 2 meet at a single place (4, 6).

2nd step: In the equation for the third line, substitute the junction point of the first two lines.

The third line’s equation is 2x + 3y = 26 —- (3) 

When we substitute values for (4, 6) in the equation (3), the result is:

2(4) + 3(6) = 26

8+ 18 = 26

26 = 26 

There are three lines that intersect each other, and the point of intersection is located right next to the third line equation. As a result, we get the concurrency of lines.

What is the concurrency of lines in Geometry?

For the purposes of a triangle, altitudes, medians, angle bisectors, and perpendicular bisectors are the four fundamental types of sets of concurrency of lines.  

Quadrilateral:

There are two lines that connect the midpoints of opposite sides and a third line that connects them. By their point of intersection, each of them is split in half. Four parallel line segments, each passing through the middle of the opposing side are contemporaneous in a cyclic quadrilateral.

Polygon:

Say, for example, that the sides of a regular polygon are all the same length. At the centre of the polygon, the diagonals connecting opposite vertices are congruent.

Circle:

At the centre of a circle, the perpendicular bisectors of all the chords are congruent. Moreover, the circumference and area bisectors of the circle are equal. Tangent lines perpendicular to their points of tangency intersect at the centre of a circle.

A Triangle’s concurrency of lines:

A triangle is a two-dimensional figure that has three sides and three angles. If lines are drawn inside a triangle, concurrent lines are possible. Regardless of the type of triangle, there are four concurrency spots. These are the four main things to keep in mind:

• Incenter:  One of the three angular bisectors (the lines that divide an angle into two equal pieces) intersects with the incenter at some point inside a particular triangle.
• Circumcenter: Three perpendicular bisectors (lines that divide an object in half at right angles) meet in the centre of a triangle.
• Centroid: Three medians (lines joining the vertices of a triangle to the midpoints of their opposite sides) cross at their centroid to form the triangle’s centroid.
• Orthocentre: The place where the three heights of a triangle meet is known as the orthocentre, or the point where the vertex meets the opposing side.

Differentiating the intersecting lines and concurrent lines: 

As we’ve already learned, if three or more line segments, rays, or lines meet at a common point, they are said to be concurring. The only thing that connects the two crossing lines is the common point where they meet.

• Concurrency of lines:
  • The intersection of three or more lines occurs at a single place.

• A point of concurrency refers to the intersection of two or more parallel lines.    

• Intersection of lines:

• In this case, only two lines cross.
• The intersection point is the intersection of two lines.

example:

How to determine the concurrency of lines in the following?

• p₁x + q₁y + r₁ = 0…………….(1)
• p2 x + q2 y + r2 = 0………(2)
• ( 2p₁ – 3p2)x + ( 2q₁ – 3q2)y + ( 2r₁ – 3r2) = 0……(3).

 Solution:

If you look at the previous three lines very closely, you’ll see that L3 is equal to the sum of the squares of the first two lines and the third line. As a result, we have three constants that are not all zeroes; therefore, pL1 + qL2 + rL3 = 0. As a result, there is a concurrency of lines in the above example.

Conclusion:

In this article, the concurrency of lines was defined and contrasted with intersecting lines. We learned the conditions for three lines to run simultaneously.

As well as parallel lines in geometry, we studied the intersection of three angular bisectors, three perpendicular bisectors, three medians, and three altitudes to form a triangle, which we called an orthocenter. We also studied parallel lines in the triangle formed by these four points: incenter, circumcenter, centroid, and orthocenter. We have also solved a few examples as per the study material notes on concurrency of lines.