• Home
  • /
  • Blog
  • /
  • What is Concurrent lines in Geometry – Definition, Conditions & Examples

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

what is concurrent lines in geometry

This post is also available in: हिन्दी (Hindi)

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.

what is concurrent lines in geometry
Is your child struggling with Maths?
frustrated-kid
We can help!
Country
  • Afghanistan 93
  • Albania 355
  • Algeria 213
  • American Samoa 1-684
  • Andorra 376
  • Angola 244
  • Anguilla 1-264
  • Antarctica 672
  • Antigua & Barbuda 1-268
  • Argentina 54
  • Armenia 374
  • Aruba 297
  • Australia 61
  • Austria 43
  • Azerbaijan 994
  • Bahamas 1-242
  • Bahrain 973
  • Bangladesh 880
  • Barbados 1-246
  • Belarus 375
  • Belgium 32
  • Belize 501
  • Benin 229
  • Bermuda 1-441
  • Bhutan 975
  • Bolivia 591
  • Bosnia 387
  • Botswana 267
  • Bouvet Island 47
  • Brazil 55
  • British Indian Ocean Territory 246
  • British Virgin Islands 1-284
  • Brunei 673
  • Bulgaria 359
  • Burkina Faso 226
  • Burundi 257
  • Cambodia 855
  • Cameroon 237
  • Canada 1
  • Cape Verde 238
  • Caribbean Netherlands 599
  • Cayman Islands 1-345
  • Central African Republic 236
  • Chad 235
  • Chile 56
  • China 86
  • Christmas Island 61
  • Cocos (Keeling) Islands 61
  • Colombia 57
  • Comoros 269
  • Congo - Brazzaville 242
  • Congo - Kinshasa 243
  • Cook Islands 682
  • Costa Rica 506
  • Croatia 385
  • Cuba 53
  • Cyprus 357
  • Czech Republic 420
  • Denmark 45
  • Djibouti 253
  • Dominica 1-767
  • Ecuador 593
  • Egypt 20
  • El Salvador 503
  • Equatorial Guinea 240
  • Eritrea 291
  • Estonia 372
  • Ethiopia 251
  • Falkland Islands 500
  • Faroe Islands 298
  • Fiji 679
  • Finland 358
  • France 33
  • French Guiana 594
  • French Polynesia 689
  • French Southern Territories 262
  • Gabon 241
  • Gambia 220
  • Georgia 995
  • Germany 49
  • Ghana 233
  • Gibraltar 350
  • Greece 30
  • Greenland 299
  • Grenada 1-473
  • Guadeloupe 590
  • Guam 1-671
  • Guatemala 502
  • Guernsey 44
  • Guinea 224
  • Guinea-Bissau 245
  • Guyana 592
  • Haiti 509
  • Heard & McDonald Islands 672
  • Honduras 504
  • Hong Kong 852
  • Hungary 36
  • Iceland 354
  • India 91
  • Indonesia 62
  • Iran 98
  • Iraq 964
  • Ireland 353
  • Isle of Man 44
  • Israel 972
  • Italy 39
  • Jamaica 1-876
  • Japan 81
  • Jersey 44
  • Jordan 962
  • Kazakhstan 7
  • Kenya 254
  • Kiribati 686
  • Kuwait 965
  • Kyrgyzstan 996
  • Laos 856
  • Latvia 371
  • Lebanon 961
  • Lesotho 266
  • Liberia 231
  • Libya 218
  • Liechtenstein 423
  • Lithuania 370
  • Luxembourg 352
  • Macau 853
  • Macedonia 389
  • Madagascar 261
  • Malawi 265
  • Malaysia 60
  • Maldives 960
  • Mali 223
  • Malta 356
  • Marshall Islands 692
  • Martinique 596
  • Mauritania 222
  • Mauritius 230
  • Mayotte 262
  • Mexico 52
  • Micronesia 691
  • Moldova 373
  • Monaco 377
  • Mongolia 976
  • Montenegro 382
  • Montserrat 1-664
  • Morocco 212
  • Mozambique 258
  • Myanmar 95
  • Namibia 264
  • Nauru 674
  • Nepal 977
  • Netherlands 31
  • New Caledonia 687
  • New Zealand 64
  • Nicaragua 505
  • Niger 227
  • Nigeria 234
  • Niue 683
  • Norfolk Island 672
  • North Korea 850
  • Northern Mariana Islands 1-670
  • Norway 47
  • Oman 968
  • Pakistan 92
  • Palau 680
  • Palestine 970
  • Panama 507
  • Papua New Guinea 675
  • Paraguay 595
  • Peru 51
  • Philippines 63
  • Pitcairn Islands 870
  • Poland 48
  • Portugal 351
  • Puerto Rico 1
  • Qatar 974
  • Romania 40
  • Russia 7
  • Rwanda 250
  • Réunion 262
  • Samoa 685
  • San Marino 378
  • Saudi Arabia 966
  • Senegal 221
  • Serbia 381 p
  • Seychelles 248
  • Sierra Leone 232
  • Singapore 65
  • Slovakia 421
  • Slovenia 386
  • Solomon Islands 677
  • Somalia 252
  • South Africa 27
  • South Georgia & South Sandwich Islands 500
  • South Korea 82
  • South Sudan 211
  • Spain 34
  • Sri Lanka 94
  • Sudan 249
  • Suriname 597
  • Svalbard & Jan Mayen 47
  • Swaziland 268
  • Sweden 46
  • Switzerland 41
  • Syria 963
  • Sao Tome and Principe 239
  • Taiwan 886
  • Tajikistan 992
  • Tanzania 255
  • Thailand 66
  • Timor-Leste 670
  • Togo 228
  • Tokelau 690
  • Tonga 676
  • Trinidad & Tobago 1-868
  • Tunisia 216
  • Turkey 90
  • Turkmenistan 993
  • Turks & Caicos Islands 1-649
  • Tuvalu 688
  • U.S. Outlying Islands
  • U.S. Virgin Islands 1-340
  • UK 44
  • US 1
  • Uganda 256
  • Ukraine 380
  • United Arab Emirates 971
  • Uruguay 598
  • Uzbekistan 998
  • Vanuatu 678
  • Vatican City 39-06
  • Venezuela 58
  • Vietnam 84
  • Wallis & Futuna 681
  • Western Sahara 212
  • Yemen 967
  • Zambia 260
  • Zimbabwe 263
Age Of Your Child
  • Less Than 6 Years
  • 6 To 10 Years
  • 11 To 16 Years
  • Greater Than 16 Years

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 LinesIntersecting 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: CodingHero - What is Concurrent lines in Geometry - Definition, Conditions & Examples Ak5qLj2vxCNQ0u7g8MXbbkaRZ2qy8t4SKW7HgUB0evvMpOrlVYxL4Jan2868ocI9Yy3RcbLEAGi5Yww5AqCD zlCiR AFapcS5ET24ompAKlm3EEX2sOqT0LwQ5Pdalc4FNMP8mh5trmtQogiduVtD59aBWfPnrnFAcpP5jOajtBoc1gGDvhfe2HdsrvLwExample: CodingHero - What is Concurrent lines in Geometry - Definition, Conditions & Examples AdZavPn0GJyT8NPONwUJOmIwqUr9gfIuyWMny18un8tyWSq6senJug4iU2zCvp1zRSYOk3b6LuUppgh2c3Mwtp8PI81ni7qhep89JMSHjb30JcaYJnFJNfXC8vXNLg x52ZsAVYuJGiDseFWccRkFbo 9K4eZZk8Ur1BxL4k2OhXS2b beofVhQS4ftMLQ

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 $0$.

what is concurrent lines in geometry

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

what is concurrent lines in geometry

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

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

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

$= 80 – 60 – 20 = 0$

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.

Types of Coordinate Systems

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

What are concurrent lines?

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.

Are parallel lines concurrent?

No, parallel lines can not be concurrent lines, because they never meet at any point. Even when parallel lines are extended indefinitely they can not be concurrent lines, since they will not have a common point at which they intersect.

How do you know if a line is concurrent?

There should be at least three lines to define a set of concurrent lines. If two lines intersect they meet at a point. If a third line also passes through this intersection point then we can say that the three lines are concurrent.

What is the difference between intersecting lines and concurrent lines?

Three lines meet at a point to form concurrent lines. The meeting point is called the ‘point of concurrence’. When two lines meet at a point, they are called intersecting lines. The meeting point of these two lines is called the ‘point of intersection’.

What are the concurrent lines of a triangle?

There are four concurrent lines for a triangle. They are medians, altitudes, angle bisectors, and perpendicular bisectors. The point of intersection is medians is called the centroid, the point of intersection of altitudes is called the orthocenter, the point of intersection of angle bisectors is called the incenter, and the point of intersection of perpendicular bisectors is called the circumcenter.

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

{"email":"Email address invalid","url":"Website address invalid","required":"Required field missing"}
>