Prove that the incentre of the triangle whose vertices are given by` A(x1, y1), B(x2, y2), C(x3,y3) is( ax1 + bx2 + cx3)/(a+b+c),( ay1+ by2 +cy3)/(a+b+c)` where a, b, and c are the sides opposite to the angles A, B and C respectively.

The co-ordinates of incentre of the triangle whose vertices are given by A(x_(1), y_(1)), B(x_(2), y_(2)), C(x_(3), y_(3)) is

The centroid of the triangle with vertices at A(x_(1),y_(1)),B(x_(2),y_(2)),c(x_(3),y_(3)) is

The centroid of the triangle formed by A(x_(1),y_(1)),B(x_(2),y_(2)),C(x_(3),y_(3)) is

Prove that the coordinates of the centre of the circle inscribed in the triangle,whose vertices are the points (x_(1),y_(1)),(x_(2),y_(2)) and (x_(3),y_(3)) are (ax_(1)+bx_(2)+cx_(3))/(a+b+c) and (ay_(1)+by_(2)+cy_(3))/(a+b+c)

A(x_(1),y_(1))B(x_(2),y_(2)) and C(x_(3),y_(3)) are the vertices and a,b and c are the sides BC,CA and AB of the triangle ABC respectively then the Coordinates of in centre are I=

if |[2a,x_1,y_1],[2b,x_2,y_2],[2c,x_3,y_3]|=(abc)/2!=0 then the area of the triangle whose vertices are (x_1/a,y_1/a),(x_2/b,y_2/b),(x_3/c,y_3/c) is (A) (1/4)abc (B) (1/8)abc (C) 1/4 (D) 1/8

A triangle has its three sides equal to a , b and c . If the coordinates of its vertices are A(x_1, y_1),B(x_2,y_2)a n dC(x_3,y_3), show that |[x_1,y_1, 2],[x_2,y_2, 2],[x_3,y_3, 2]|^2=(a+b+c)(b+c-a)(c+a-b)(a+b-c)

If A(x_(1),y_(1)),B(x_(2),y_(2)) and C(x_(3),y_(3)) are the vertices of a triangle then excentre with respect to B is

If the points A(3,4),B(x_1 ,y _1) and C(x_2 ,y_2) are such that both 3 , x_1, x_2 and 4, y_1 ,y_2 are in AP, then (i) A , B , C are vertices of an isosceles triangle (ii) A , B , C are collinear points (iii) A , B , C are vertices of a right angle triangle (iv) A , B , C are vertices of a scalene triangle (v) A , B , C are vertices of a equilateral triangle