Two lines in a plane can either be parallel or intersecting. Three lines in a plane may (i) be parallel to each other (ii) intersect each other in exactly one point (iii) intersect each other in two points (iv) 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.
Angle Bisectors of Triangle
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.
AD, BE and CF are three angle bisectors of ΔABC which passes through same point I. I 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 i.e. IL = IM = IN.
If we take I as centre and IL or IM or IN as radius and draw a circle then the circle is called Incircle of the triangle.
Perpendicular Bisectors of Sides of Triangle
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.
Circumcentre will be
- In the interior of the triangle for an a acute triangle.
- On the hypotenuse of a right angle.
- in the exterior of the triangle for an obtuse triangle.
All the three perpendicular bisectors of a triangle pass through O and it is called circumcentre which is equidistant from vertices A, B and C.
If we mark O as centre and OA or OB or OC as radius and draw a circle. The circle passes through A, B and C of the triangle called Circumcircle of the triangle.
Altitudes of Triangle
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.
- In the exterior of the triangle for an obtuse triangle.
Medians of Triangle
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 the 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.
The three medians AD, BE and CF are concurrent at G.
AG = 2GD, BG = 2GE and CG = 2GF
Example: In an isosceles triangle, show that the bisector of the angle formed by the equal sides is also a perpendicular bisector, an altitude and a median of the triangle.
Example: In an equilateral triangle, show that the three angle bisectors are also the three perpendicular bisectors of sides, three altitudes and the three medians of the triangle.
Example: Find the circumradius of circumcircle and inradius of incircle of an equilateral triangle of side a.