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How to Prove Three Lines are Concurrent?

TopConcurrent lines are defined as the Set of lines passing through a same Point. That is, it is defined as a set of lines which intersects each other at a single point. The point at which two or more lines intersect each other is known as point of concurrency or concurrent point. We may come across a question that, How to prove three lines are concurrent? for this, we need to find whether the lines are intersecting or not. Suppose if three lines are intersecting then the point of Intersection must be same for all three lines. If the point of intersection is same then we can conclude that all three lines are concurrent to each other.


Triangle is a good example to understand concurrent lines as it contains four Sets of concurrent lines that are, angle bisector, median, altitudes and perpendicular bisectors. First we need to understand about all these terms.

Angle bisector: it is defined as the line which bisects angles of triangle into two equal halves. There are three angles in a triangle; therefore there will be three angle bisectors. And all three angle bisectors will meet or intersect at a same point. Thus, it is a set of concurrent lines.
Median: it is defined as the line which bisects the sides of triangle into two equal halves. It is projected Onto the sides from opposite angle of the respective side. Thus all three medians of a triangle meet at same point forming the set of concurrent sides.
Altitudes: altitudes are defined as the lines perpendicular to the sides of triangle. This is also projected on the sides from opposite respective angles. More than three lines can also be concurrent if they intersect on the same point. Thus in the above discussion we get to know about how to prove three lines are concurrent.