What Are Concurrent Lines Definition Properties Formula and Solved Examples
It is commonly known that two non-parallel intersect at one point. If a third line is formed passing through one common point or intersecting each other at one common point, then these straight lines are termed as concurrent lines. The word ‘concurrent ‘ means something that occurs at the same time or same point.
By Eculid’s Lemma, it is stated that two lines have a maximum one common point of intersection. In the figure given below, we can see that lines are meeting each other at point P. When three or more lines intersect together exactly at one single point in a plane then they are termed as concurrent lines. The point where three or more lines meet each other is termed as the point of concurrency.
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In this article, we will discuss concurrent lines, concurrent lines definition, concurrent line segments and rays, differences between concurrent lines and intersecting lines etc.
Concurrent Line Definition
A set of three or more lines are termed as concurrent when passing through one common point or coincide exactly at one common point. The common point where all the lines intersect or coincide is known as the point of concurrency. In the figure given below, the line shown in blue, orange, and black is passing through the point O. Hence, all three lines are concurrent to each other.
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Concurrent Line Segments and Rays
When three or more line segments are intersecting each other at one single point then these lines are determined as concurrent line segments. In the figure given below, AB ,CD and EF are three line segments intersecting each other at point O. Hence, it can be said that concurrency can also be applied to line segments.
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When three or more rays in a two - dimensional plane intersect each other at one single point, then they are termed as concurrent rays. The common point where all the rays meet each other is termed as the point of concurrency for all the rays. In the figure given below, three rays PQ, RS and MN which are meeting each other at point O are concurrent with each other.
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Point of Concurrency
When three or more lines pass through a same point they are called concurrent lines. The point where they all meet or intersect is called the point of concurrency. For example if line A, line B and line C are concurrent lines which meet at a point say Z, then Z would be the point of concurrency where the three lines meet each other.
Concurrent Lines of a Triangle
A triangle is a 2D shape with three sides and three angles. There can be concurrent lines in a triangle if line segments are drawn inside a triangle. A triangle has four different concurrency points irrespective of the type of the triangle. These four points are-
Incenter- This is a point of intersection of the three angular bisectors (lines dividing the angles into two equal parts) inside a given triangle.
Circumcenter- This is a point of intersection of the three perpendicular bisectors ( lines that divide a given line into two equal parts at right angle) and inside a triangle.
Centroid- This is a point of intersection of the three medians (line joining the vertex to the midpoint of the opposite side) of a given triangle.
Orthocentre- This is the point of intersection of the three altitudes (line joining the vertex to the opposite side and is perpendicular) of a triangle.
Difference Between Concurrent Lines and Intersecting Lines
As we know that if three or more lines, line segments, or rays meet each other at one common point then they are said to be in concurrency. But in the case of intersecting lines, there are only two lines, line segments or rays that meet each other at one common point.
Below are some points which show differences between concurrent lines and intersecting lines in tabulated form
Quiz Time
What is Represented as the Point of Concurrency for the Median of a Triangle?
Circumcentre
Centroid
Incener
Orthocenter
2. Three or More Lines Are Considered as Concurrent I That Pass Through
Same line
Same point
Same plane
None of the above
Solved Examples
1. Show That the Three Lines 2p - 4q + 5 = 0, 7p - 8q + 5 and 4p + 5q = 45 Are Concurrent Lines and Also Determine the Point of Concurrency.
Solution:
Let,
2p - 4q + 5 = 0 ……… (1)
7p - 8q + 5 = 0 ……….(2)
4p + 5q = 45 …………(3)
Let us use the substitution method and solve equations 1 and 2 given above.
2p + 5 = 4q
7p - 8q + 5 = 0 or 7p - 2(4q) + 5 = 0
Now substitute 4q = 3p + 5……..(3)
7p - 2(3p + 5) + 5 = 0
7p - 6p - 10 + 5 = 0
p - 5 =0
p = 5
Hence, the point of intersection of lines 1 and 2 is (5,5).
Now , let us examine whether the third line satisfies the point (5,5).
i.e. 4p + 5q = 45
(3) → 4(5) + 5(5) = 45
20 + 25 = 45
45 = 45
Hence, all three given lines are passing through the point (5,5) and they are said to be concurrent lines. And the point of concurrency is (5,5).
2. Verify, If the Following Lines are Concurrent
p1x + q1y + r1 = 0…………….(1)
p2 x + q2 y + r2 = 0………(2)
( 2p1 - 3p2)x + ( 2q1 - 3q2)y + ( 2r1 - 3r2) = 0……(3).
Solution:
If we carefully see the above three lines, we will notice that if the given lines are represented by L1 , L2 and L3, then we have L3 - 2L1 + 3L2 . Hence, we have three constants, not all zero such that pL1 + qL2 + rL3 = 0. Therefore, the given lines are concurrent.
FAQs on Concurrent Lines in Geometry with Definition and Explanation
1. What are concurrent lines in geometry?
Concurrent lines are three or more lines that intersect at a single common point. This common point is called the point of concurrency.
- If two lines intersect, they meet at one point but are not considered concurrent unless a third line also passes through that same point.
- Concurrency is commonly studied in triangles, where special lines meet at specific points.
- Example: The three medians of a triangle intersect at one point, making them concurrent.
2. What is the point of concurrency?
The point of concurrency is the single point where three or more lines intersect. In geometry, different sets of concurrent lines have special names for their concurrency point.
- Medians meet at the centroid.
- Angle bisectors meet at the incenter.
- Perpendicular bisectors meet at the circumcenter.
- Altitudes meet at the orthocenter.
3. Are the medians of a triangle concurrent?
Yes, the three medians of a triangle are concurrent at the centroid. A median is a line drawn from a vertex to the midpoint of the opposite side.
- All three medians intersect at one point called the centroid.
- The centroid divides each median in the ratio 2:1 (vertex to centroid is twice centroid to midpoint).
- The centroid is always inside the triangle.
4. How do you prove that three lines are concurrent?
To prove three lines are concurrent, you must show they intersect at the same single point. This can be done using algebra or geometric theorems.
- Algebraic method: Solve two line equations to find their intersection point, then verify the third line also passes through that point.
- Geometric method: Use known theorems (e.g., medians meet at centroid).
5. What is the difference between concurrent lines and parallel lines?
Concurrent lines intersect at one common point, while parallel lines never intersect. The key differences are:
- Concurrent lines: Meet at a single shared point.
- Parallel lines: Remain equidistant and do not meet.
- Concurrent lines form angles at the intersection point, while parallel lines do not intersect at all.
6. What are examples of concurrent lines in a triangle?
In a triangle, several important sets of lines are concurrent. These include:
- Medians → meet at the centroid.
- Angle bisectors → meet at the incenter.
- Perpendicular bisectors → meet at the circumcenter.
- Altitudes → meet at the orthocenter.
7. Can two lines be called concurrent?
No, two lines alone are not called concurrent because concurrency requires at least three lines intersecting at one point. While two lines can intersect, the term “concurrent” is used only when three or more lines share the same intersection point.
8. How do you find the point of concurrency using equations?
To find the point of concurrency using equations, solve the system of line equations to get their common intersection point.
- Step 1: Solve any two line equations simultaneously.
- Step 2: Find their intersection point (x, y).
- Step 3: Substitute (x, y) into the third equation to verify it satisfies it.
9. Why are perpendicular bisectors concurrent?
The perpendicular bisectors of a triangle are concurrent because they intersect at the circumcenter, which is equidistant from all three vertices. This happens because:
- Each perpendicular bisector contains points equidistant from the endpoints of a side.
- Their intersection point is equidistant from all three vertices.
- This point is the center of the circumscribed circle (circumcircle).
10. What is a real-life example of concurrent lines?
A real-life example of concurrent lines is the spokes of a bicycle wheel, which meet at the center hub. Other examples include:
- Roads meeting at a roundabout.
- Light rays focused at a single point.
- Structural beams meeting at a joint.
