If `A(x_1, y_1), B(x_2, y_2), C(x_3, y_3)` are the vertices of a `DeltaABC and (x, y)` be a point on the median through `A`. Show that : `|(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`

If A(x_(1), y_(1)), B(x_(2), y_(2)) and C (x_(3), y_(3)) are the vertices of a Delta ABC and (x, y) be a point on the internal bisector of angle A, then prove that b|(x,y,1),(x_(1),y_(1),1),(x_(2),y_(2),1)|+c|(x,y,1),(x_(1),y_(1),1),(x_(3),y_(3),1)|=0 where, AC = b and AB = c.

If A(x_1, y_1),B(x_2, y_2) and C(x_3,y_3) are vertices of an equilateral triangle whose each side is equal to a , then prove that |[x_1,y_1, 2],[x_2,y_2, 2],[x_3,y_3, 2]|^2=3a^4

If A (x_(1), y_(1)), B (x_(2), y_(2)) and C (x_(3), y_(3)) are vertices of an equilateral triangle whose each side is equal to a, then prove that |(x_(1),y_(1),2),(x_(2),y_(2),2),(x_(3),y_(3),2)| is equal to

If A(x_1, y_1),B(x_2, y_2),C(x_3, y_3) are the vertices of a triangle, then the equation |x y1x_1y_1 1x_2y_3 1|+|x y1x_1y_1 1x_3y_3 1|=0 represents (a)the median through A (b)the altitude through A (c)the perpendicular bisector of B C (d)the line joining the centroid with a vertex

If P(x_1,y_1),Q(x_2,y_2) and R(x_3, y_3) are three points on y^2 =4ax and the normal at PQ and R meet at a point, then the value of (x_1-x_2)/(y_3)+(x_2-x_3)/(y_1)+(x_3-x_1)/(y_2)=

If (x_1,y_1),(x_2,y_2)" and "(x_3,y_3) are the feet of the three normals drawn from a point to the parabola y^2=4ax , then (x_1-x_2)/(y_3)+(x_2-x_3)/(y_1)+(x_3-x_1)/(y_2) is equal to

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)

If the coordinates of the vertices of an equilateral triangle with sides of length a are (x_(1),y_(1)),(x_(2),y_(2)) and (x_(3),y_(3)) then |(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1)|^2=(3a^(4))/4

Find the coordinates of the centroid of the triangle whose vertices are (x_1,y_1,z_1) , (x_2,y_2,z_2) and (x_3,y_3,z_3) .

STATEMENT-1: If three points (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)) are collinear, then |{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|=0 STATEMENT-2: If |{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|=0 then the points (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)) will be collinear. STATEMENT-3: If lines a_(1)x+b_(1)y+c_(1)=0,a_(2)=0and a_(3)x+b_(3)y+c_(3)=0 are concurrent then |{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=0