# Centroid and Incenter

The point which divides a median of a triangle in the ratio 2 : 1 is called
the centroid of the triangle. Thus, if AD is a median of the triangle ABC and
G is its centroid, then

**AG/GD = 2/1**

By section formula, the co-ordinates of G are

The symmetry of the co-ordinates of G shows that it also lies on the medians
through B and C. Hence the medians of a triangle are concurrent.

**Incenter of a triangle**

The point of the intersection of any two internal bisectors of the angles
of a triangle is called the incenter of the triangle. It is usually denoted by I.

If the internal bisector of
A of a
ABC meets the side BC in D, then

**BD/DC = AB/AC**

By section formula, the co-ordinates of I are

The symmetry of the co-ordinates of I shows that it also lies on the internal
bisector of C.
Hence the internal bisectors of the angles of a triangle are concurrent.

## Illustrative Examples

### Example

Find the co-ordinates of the incenter of the triangle whose vertices are(-2,4), (5,5) and (4,-2).

### Solution

Let A (-2,4), B (5,5) and C (4,-2) be the vertices of the given triangle ABC, then

a = | BC| = [(4 -5)² +(-2 -5)²] =[1
+49] = 50 = 52,

b = |CA| = [(4 -2)² +(-2 -4)²] = [36
+36] = 72 = 62 and

c = |AB| = [(5 -2)² +(5 -4)²] = [49
+1] = 50 = 52.

The co-ordinates of the incenter of ABC are

## Exercise

- Find the centroid of the triangle whose vertices are (-1,4), (2,7) and (-4,-3).
- Find the point of intersection of the medians of the triangle whose
vertices are (3,-5), (-7,4) and (10,-2). [
**Hint.**Find centroid.] - Find the third vertex of a triangle if two of its vertices are (3,-5) and (-7,4), and the medians meet at (2,-1).
- Find the centroid of the triangle ABC whose vertices are A(9,2), B(1,10) and C(-7,-6). Find the co-ordinates of the middle points of its sides and hence find the centroid of the triangle formed by joining these middle points. Do the two triangles have same centroid?
- If (-1,5), (2,3) and (-7,9) are the middle points of the sides of a triangle, find the co-ordinates of the centroid of the triangle.
- If A(1, 5), B (-2,1) and C(4,1) are the vertices of ABC, and internal bisector of A meets side BC at D, find |AD|. Also find the incenter of ABC.
- Find the co-ordinates of the center of the circle inscribed in a triangle whose angular points are (-36,7), (20,7) and (0,-8).

## Answers

**1.**(-1,8/3)

**2.**(2,-1)

**3.**(10,-2)

**4.**(1,2); mid-points of BC, CA, AB are (-3,2), (1,-2), (5,6); (1,2); Yes

**5.**(-2,17/3)

**6.**4 units; (1,5/2)

**7.**(-1,0)