Median, Altitude, Incentre and Circumcircle

Median

 The median of a triangle corresponding to any side is the segment joining the midpoint of that side with the opposite vertex.

            In the given figure, D, E, F are the respective midpoints of sides BC, CA and AB of triangle ABC.

∴    AD is the median, corresponding to side BC. BE is the median, corresponding to side CA. CF is the median, corresponding to side AB.

The medians of a triangle are concurrent, i.e., they interest each other at the same point.

Cendroid: The point of intersection of all the three medians of a triangle is called its centroid.

In the above figure, the medians AD, BE and CF of triangle ABC interest at the point G.

∴      G is the centroid of triangle ABC.

Altitudes

The altitude of a triangle corresponding to any side is the length of perpendicular drawn from the opposite vertex to that side.

The side on which the perpendicular is being drawn, is called its base.

       In the given figure, AL ⟘ BC; BM ⟘ CA and CN ⟘ AB.

∴      AL is the altitude, corresponding to base BC.

BM is the altitude, corresponding to base CA.

CN is the altitude, corresponding to base AB.

A triangle has three altitudes.

The altitudes of a triangle are concurrent.

Orthocentre:  The point of intersection of all the three altitudes of a triangle is called its orthocentre.

                 In the above figure, the three altitudes AL, BM and CN of △ ABC interest at a point H.

∴        H is the orthocentre of △ ABC.

Incentre of a Triangle

The point of intersection of the internal bisector of the angles of a triangle is called its incentre.

                 In the given figure, the internal bisectors of the angles of △ ABC intersect at I.

∴         I is the incentre of △ ABC, Let ID ⊥ BC.

Then, a circle with centre I and radius ID is called the incircle of △ ABC.

Circumference of a Triangle

The point of intersection of the perpendicular bisectors of the sides of a triangle is called its circumference.

            In the given figure, the right bisectors of the sides of △ ABC intersect at O.

∴         O is the circumference of △ ABC.

With O as centre and radius equal to OA = OB = OC, we draw a circle passing through the vertices of the given triangle.

This circle is called the circumference of △ ABC.