If `A(x_(1),y_(1)),B(x_(2),y_(2))andC(x_(3),y_(3))` are vertices of on equilateral triangle with each side equal to a units, than 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 the triangle then show that:'

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 the vertices of a triangle then excentre with respect to B is

Suppose (a_(1),b_(1)),(a_(2),b_(2)) and (a_(3),b_(3)) are vertices of an equilateral triangle whose each side is of length2.1 units, then ||(x_(1),y_(1),8sqrt(3)),(x_(2),y_(2),8sqrt(3)),(x_(3),y_(3),8sqrt(3))||= __________

An equilateral triangle has each of its sides of length 6 cm. If (x_(1),y_(1)), (x_(2),y_(2)) & (x_(3),y_(3)) are the verticles, then the value of the determinant |{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|^(2) is equal to :

If (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)) are vertices of equilateral triangle such that (x_(1)-2)^(2)+(y_(1)-3)^(2)=(x_(2)-2)^(2)+(y_(2)-3)^(2)=(x_(3)-2)^(2)+(y_(3)-3)^(2) then

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

If (x_(1),y_(1),z_(1)),(x_(2),y_(2),z_(2)) and (x_(3),y_(3),z_(3)) are the vertices of an equilateral triangle such that (x_(1)-2)^(2)+(y_(1)-3)^(2)+(z_(1)-4)^(2)=(x_(2)-2)^(2)+(y_(2)-3)^(2)+(z_(2)-4)^(2) =(x_(3)-2)^(2)+(y_(3)-3)^(2)+(z_(3)-4)^(2) then sum x_(1)+2sum y_(1)+3sum z_(1)