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Points of a Triangle

Before we move ahead to discuss different points inside a triangle, we need to be very clear about some of the basic definitions.

Basic Definitions

Altitude (or height)
Description: P-326-4.tif
AF, CD and BE are the altitudes

Property: The perpendicular drawn from the opposite vertex of a side in a triangle is called an altitude of the triangle. There are three altitudes in a triangle.

Description: 23432.png
i.e., AE, CD and BF are the medians (BE = CE = AD = BD, AF = CF)

Property: The line segment joining the mid-point of a side to the vertex opposite to the side is called a median. There are three medians in a triangle. A median bisects the area of the triangle. Area (ABE) = Area (AEC) Description: 6953.png Area (∆ABC) etc.

Angle bisector
Description: 23439.png
AE, CD and BF are the angle bisectors

Property: A line segment which originates from a vertex and bisects the same angle is called an angle bisector.(âBAE = CAE Description: 6938.png BAC) etc.

Perpendicular bisector
Description: 23446.png
DO, EO and FO are the perpendicular bisectors

Property: A line segment which bisects a side perpendicularly (i.e., at right angle) is called a perpendicular bisector of a side of triangle. All points on the perpendicular bisector of a line are equidistant from the ends of the line.


Circumcentre is the point of intersection of the three per­pendicular bisectors of a triangle. The circumcentre of a triangle is equidistant from its vertices and the distance of the circumcentre from each of the three vertices is called circumradius (R) of the triangle. These perpen­dicular bisectors are different from altitudes, which are perpendiculars but not necessarily bisectors of the side.

The circle drawn with the circumcentre as the centre and circumradius as the radius is called the circumcircle of the triangle and it passes through all the three vertices of the triangle.

The circumcentre of a right-angled triangle is the mid-point of the hypotenuse of a right- angled triangle.
Description: 23453.png
AB = c, BC = a, AC = b
The process to find the circumradius (R) For any triangle Description: 6897.png where ab and c are the three sides and A = area of a triangle.
For equilateral triangle Description: 6890.png

Positioning of the Circumcentre

  • If the triangle is acute-angled triangle, then the circumcentre will lie inside the triangle.
  • If the triangle is obtuse-angled triangle, then the circumcentre will lie outside the triangle.
  • If the triangle is a right-angled triangle, then the circumcentre will lie on the mid-point of the hypotenuse. This can be seen through the following diagram:
Description: P-328-1.tif
Here D is the circumcentre. So, AD = CD = BD


Incentre is the point of intersection of the internal bisectors of the three angles of a triangle. The incentre is equidistant from the three sides of the triangle i.e., the perpendiculars drawn from the incentre to the three sides are equal in length and are called the inradius of the triangle.
The circle drawn with incentre as the centre and inradius as the radius is called the incircle of the triangle and it touches all the three sides from the inside.
Description: P-328-2.tif
AB = c, BC = a, CA = b
To find inradius (r)
For any triangle Description: 6866.png where
A = Area of triangle and
S = Semi-perimeter of the triangle Description: 6858.png
For equilateral triangle Description: 6851.png
BIC = 90° + A/2
Important derivation – In a right-angled triangle, Inradius = Semi-perimeter – Length of Hypotenuse.

Euler’s formula for inradius and circumradius of a triangle
 Let O and I be the Circumcentre and Incentre of a triangle with circumradius R and inradius r. Let d be the distance between O and I. Then
d2 = R(R − 2r)
From this theorem, we obtain the inequality r ≥ 2r. This is known as Euler’s Inequality.


Centroid is the point of intersection of the three medians of a triangle. The centroid divides each of the medians in the ratio 2 : 1, the part of the median towards the vertex being twice in length to the part towards the side.
Description: P-328-3.tif
​​Description: 6836.png

Median divides the triangle into two equal parts of the same area.


The point of concurrency of the altitudes is known as the orthocentre. Summarizing the above discussion regarding the points of the triangle.

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