Concurrent Lines

Two lines can either be parallel or intersecting. Three lines may be parallel to each other, intersect each other in exactly one point, intersect each other in two points (transversal), or intersect each other at most in three points.

Three or more lines in a plane which intersect each other in exactly one point or pass through the same point are called concurrent lines and the common point is called the point of concurrency.

1. Angle Bisectors

A line which bisects an angle of a triangle is called an angle bisector of the triangle. A triangle has three angle bisectors in it. Angle bisectors of a triangle pass through the same point, I. It is called incentre of the triangle. Incentre always lies in the interior of the triangle and at the same distance from the three sides of the triangle.

2. Perpendicular Bisectors

A line which bisects a side of a triangle at right angle is called the perpendicular bisector of the side. The three perpendicular bisectors of the sides of a triangle pass through the same point. The point of concurrency O is called the circumcentre of the triangle. The circumcentre is equidistant from vertices A, B and C. Circumcentre will be in the interior of the triangle for an a acute triangle, on the hypotenuse of a right angle, and in the exterior of the triangle for an obtuse triangle.

3. Altitudes

Perpendicular drawn from a vertex of a triangle on the opposite side is called its altitude. In a triangle the three altitudes pass through the same point and the point of concurrency is called the orthocentre of the triangle. Orthocentre will be in the interior of the triangle for an acute triangle, at the vertex containing the right angle for a right triangle, and in the exterior of the triangle for an obtuse triangle.

4. Medians

A line joining a vertex to the mid point of the opposite side of a triangle is called its median. All the three medians pass through the same point. The point of concurrency G is called centroid of the triangle. Centroid divides each of the medians in the ratio 2:1.

In an isosceles triangle, bisector of the angle formed by the equal sides is also a perpendicular bisector, an altitude and a median of the triangle.

In an equilateral triangle the angle bisectors are also the perpendicular bisectors of the sides, altitudes and medians of the triangle.