Concurrent lines
Three or more lines are said to be concurrent if they all pass through the same point. The common point is called the point of concurrency.
Terms related to a triangle
Median
The line joining the vertex to the midpoint of the opposite side is called a median of a triangle. A triangle has 3 medians. The 3 medians in a triangle are concurrent.
Altitude
The perpendicular drawn from a vertex to the line containing the opposite side is called an altitude of the triangle. A triangle has 3 altitudes.
Perpendicular Bisector
The locus of a point equidistant from two given points is the perpendicular bisector of the line segment joining the two points. In a triangle there will be 3 perpendicular bisectors and they are concurrent.
Angle Bisector
The locus of a point equidistant from two intersecting lines is the pair of lines bisecting the angles formed by the given lines. The 3 angle bisectors of triangle are concurrent.
Incentre and Incircle
- The point of concurrency of the angle bisectors of a triangle is called the incentre of the triangle. The point 'I' is called the Incentre of a triangle.
- If a circle is drawn with center 'I' and radius as the length of perpendicular from 'I' on a side, then this circle will touch all the sides of the triangle. This circle is called the Incircle of the triangle.
Circumcentre and Circumcircle
- The point of concurrency of the perpendicular bisectors of the sides of a triangle is called the circumcentre of the triangle.
- The point 'O' is called the circumcentre of a triangle.
- If 'O' is the circumcentre of a triangle then a circle with 'O' as center and OA as radius (to any side 'A' of the triangle ABC) passes through the vertices of the triangle, and it is known as the circumcircle of the triangle.
Centroid
- The point of concurrency of the three medians of a triangle is called the centroid of the triangle.
- The centroid of a triangle divides each one of its medians in the ratio 2:1.
- G is the centroid of the triangle.
Orthocentre
- The point of concurrency of the three altitudes of a triangle is called its orthocentre.
- H is called the orthocentre of the triangle.
Properties on concurrent lines in a triangle
- The angle bisectors of a triangle pass through the same point.
The angle bisectors of a triangle pass through the same point and hence they are concurrent.
- The perpendicular bisectors of the sides of a triangle pass through the same point.
- The altitudes of a triangle pass through the same point.
