The coordinatse of the vertices A, B and C of the triangle ABC taken in anticlockwise order are respectively `(x_4 , y_r ), r = 1, 2, 3`. Prove that the angle `A` is acute or obtuse according as : `(x_1 - x_2) (x_1 - x_3) + (y_1 - y_2) (y_1 - y_3) gt 0 or lt0`.

Let (x_r,y_r) r = 1 ,2,3 are the coordinates of the vertices of a triangle ABC. If D is the point onit in the ratio of 1 : 2. reckoning from the vertex B, prove that the equation of the line AD is 2|(x,y,1),(x_1,y_1,1),(x_2,y_2,1)|+|(x,y,1), (x_1, y_1, 1), (x_3,y_3,1)|=0. Also find the equation of the line AE in the similar form where E is the harmonic conjugate of D wrt the points B and C.

If the points (x_1, y_1), (x_2, y_2) and (x_3, y_3) be collinear, show that: (y_2 - y_3)/(x_2 x_3) + (y_3 - y_1)/(x_3 x_2) + (y_1 - y_2)/(x_1 x_2) = 0

If x, y gt 0 , then prove that the triangle whose sides are given by 3x + 4y, 4x + 3y, and 5x + 5y units is obtuse angled.

If the co-ordinates of the vertices of an equilateral triangle with sides of length a are (x_1,y_1), (x_2, y_2), (x_3, y_3), then |[x_1,y_1,1],[x_2,y_2,1],[x_3,y_3,1]|=(3a^4)/4

Find the coordinates of the centroid of a triangle having vertices P(x_1,y_1,z_1),Q(x_2,y_2,z_2) and R(x_3,y_3,z_3)

A (3,4 ), B (-3, 0) and C (7, -4) are the vertices of a triangle. Show that the line joining the mid-points D (x_1, y_1), E (x_2, y_2) and F (x, y) are collinear. Prove that (x-x_1) (y_2 - y_1) = (x_2 - x_1) (y-y_1)

Let ABC is a triangle whose vertices are A(1, –1), B(0, 2), C(x', y') and area of triangleABC is 5 and C(x', y') lie on 3x + y – 4lambda = 0 , then

In a triangle ABC.if A(1,2) and the internal angle bisetors through B and C are y=x and y=-2x, then the in radius r of Delta ABC is :

If x_(1),x_(2),y_(1),y_(2)in R if ^(@)0