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.
Find the co-ordinates of the incenter of the triangle whose vertices are(-2,4), (5,5) and (4,-2).
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
- 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).
Answers1. (-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)