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Concurrent lines are three or more lines that pass through the same point in a plane. A point of intersection is formed when two non-parallel lines cross each other. These three lines are considered to be concurrent when another line passes through the point of junction formed by the first two lines.
The “Place of Concurrency” is the point where all three lines intersect. Three altitudes drawn on a triangle, for example, intersect at a location called the orthocentre. It’s worth noting that only non-parallel lines can have a point of concurrence because they run forever and intersect at some point.
A few examples include a circle’s diameter and its centre. The line segments connecting the midpoints of opposite sides and the diagonals are contemporaneous in quadrilaterals. The three lines in the diagram below intersect at point P. All three lines are running at the same time.
Difference between concurrent lines and intersecting lines:
As previously stated, any three lines, line segments, or rays that have a single point of the junction are said to be in concurrency. Intersecting lines, on the other hand, consist of only two lines, line segments, or rays that cross each other. The differences can be written in a table.
Concurrent lines | Interesting lines |
A single point is crossed by three or more lines. | There are only two lines that cross each other. |
A point of concurrency is the intersection of these lines at a single location. | The point of intersection is defined as the junction of two lines. |
Concurrent lines geometry:
Triangle:
Altitudes, angle bisectors, medians, and perpendicular bisectors are the four primary types of concurrent lines in a triangle.
Quadrilateral:
The line segment connecting the midpoints of the diagonals and the two segments connecting the midpoints of opposite sides are both contemporaneous. Their point of intersection cuts them all in half.
Four line segments, each perpendicular to one side and crossing through the midpoint of the other side, are contemporaneous in a cyclic quadrilateral.
Polygon:
Assume an even number of sides for a regular polygon. In this situation, the diagonals connecting opposite vertices are parallel to the polygon’s centre.
Circle:
At the circle’s centre, the perpendicular bisectors of all of the chords are parallel.
A circle’s perimeter and area bisectors are both diameters, and they intersect at the circle’s centre.
At the point of tangency, the lines perpendicular to the tangents to a circle are contemporaneous.
Concurrent lines in triangles:
In a triangle, the concurrent lines are:
Altitudes
Medians
Angle bisectors
Perpendicular bisectors
Altitudes | At a common point, the three heights of the triangle from all three vertices intersect. The orthocenter is the location where the altitudes cross. |
Medians | The centroid is the intersection of three triangle medians that divide the opposite side into equal pieces and intersect at a single location. |
Angle bisectors | Angle bisectors are rays that cut the angle in half from each vertex and meet at a single point. In this case, the point is referred to as incenter. |
Perpendicular bisectors | The perpendicular bisectors are lines that travel through a single point and cross the opposite sides at 90 degree angles. The circumcenter is the name given to this place. |
Example of concurrent lines:
Any median (which must be a bisector of the triangle’s area) occurs simultaneously with two additional area bisectors, each parallel to a side.
A triangle cleaver is a line segment that bisects the triangle’s perimeter and has one terminal at the midpoint of one of its three sides. At the middle of the Spieker circle, which is the incircle of the medial triangle, the three cleavers meet.
Each triangle has one, two, or three of these lines that divide the triangle’s area and perimeter in half. If there are three of them, they will all agree on the incenter.
The point of convergence of the Euler lines of four triangles: the triangle in question, and the three triangles that share two vertices with it and have their incenter as the other vertex.
Conclusion:
If two lines in a plane or higher-dimensional space intersect at a single point, they are said to be contemporaneous. They are the polar opposite of parallel lines. Concurrent lines are three or more lines that pass through the same point in a plane. A point of intersection is formed when two non-parallel lines cross each other. These three lines are considered to be concurrent when another line passes through the point of junction formed by the first two lines.
The “Place of Concurrency” is the point where all three lines intersect. Three altitudes drawn on a triangle, for example, intersect at a location called the orthocentre. A few examples include a circle’s diameter and its centre. The line segments connecting the midpoints of opposite sides and the diagonals are contemporaneous in quadrilaterals.
As previously stated, any three lines, line segments, or rays that have a single point of the junction are said to be in concurrency. Intersecting lines, on the other hand, consist of only two lines, line segments, or rays that cross each other.
Each triangle has one, two, or three of these lines that divide the triangle’s area and perimeter in half. If there are three of them, they will all agree on the incenter.
