(i) Median and centroid: A line joining the mid-point of a side of a triangle to its opposite vertex is called the median. D,E,F are the mid-points of the sides QR, PR and PQ respectively of a ΔPQR. Then, PD, QE and RF are the medians of ΔPQR.
• The point of concurrency of the three medians of a triangle is called centroid.
• The centroid of a triangle divides each median in the ratio 2:1, i.e.,
PG : GD= QG : GE= RG : GF = 2:1
• A median divides a triangle into two parts of equal area.
(ii) Perpendicular bisector and circumcentre: Perpendicular bisector to any side is the line that is perpendicular to that side and passes through its mid-point. Perpendicular bisectors need not pass through the opposite vertex.
• The point of intersection of the three perpendicular bisectors of a triangle is called its circumcentre.
• The circumcentre of a triangle is equidistant from its three vertices.
If we draw a circle with circumcentre as the centre and the distance of any vertex from the circumcentre as radius, the circle passes through all the three vertices and the circle is called circumcircle.
Note. The circumcentre can be inside or outside the circle.
• Circumcentre of a right angled triangle is the mid-point of the hypotenuse
(iii) Angle bisector and in-centre:
• The point of intersection of the three angle bisectors of a triangle is called its in-centre.
• The in-centre always lies inside the triangle.
• It is always equidistant from the sides of a triangle.
• The circle drawn with incentre as centre and touching all the three sides of a triangle is called in-circle.
(iv) Altitude and ortho-centre:
The perpendicular drawn from the vertex of a triangle to the opposite side is called an altitude.
• The point of intersection of the three altitudes of a triangle is called ortho-centre, which can lie inside or outside the triangle.
Note. • For an isosceles triangle, the median drawn from a vertex to the opposite side is also the perpendicular bisector of that side.
• In an equilateral triangle, the median, angle bisector, altitude and perpendicular bisector of sides are all represented by the same straight line.
• The circumcentre, centroid, orthocentre and incentre all coincide in an equilateral tria
