Introduction
Recall that if two straight lines are not parallel, they meet at a point when produced.
Concurrent Line Segments Associated with a Triangle
Recall a line segment joining the vertex to the mid-point of the opposite side of a triangle.
Concurrency of the Angle Bisectors of Angles of a Triangle
Draw a triangle ABC. Draw its angle bisectors. The angle bisectors of a triangle are concurrent.
Concurrency of the Angle Bisectors of a Triangle
Draw a triangle ABC. Draw its angle bisectors. The angle bisectors pass through a point.
Concurrency of the Perpendicular Bisectors of the Sides of a Triangle
Draw a triangle ABC. Draw the perpendicular bisectors of its sides. The perpendicular bisectors of the sides of a triangle are concurrent (pass through the same point).
Theorem1
To determine the incentre of a triangle, it is just sufficient to find the point of intersection of its two angles. The third angle bisector is bound to pass through it by virtue of the below theorem.
Theorem2
To determine the circumcentre of a triangle, it is just sufficient to find the point of intersection of any two perpendicular bisectors of the sides of a triangle.>
Theorem3
To locate orthocentre it is sufficient to draw altitudes of any two sides of a triangle. The third altitude will then automatically pass through it.
Theorem4
The medians of a triangle pass through the same point which divides each of the medians in the ratio 2:1.
Summary
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.
