For what value of `alpha` ,the points `(1,2,3),(2,3,1)` and `(3,1,alpha)` form an equilateral triangle?

For what values of alpha ,the points (0,-1,0),(2,1,-1),(1,1,1) and (3,3,alpha) are coplanar

Show that the points (1,-2,-3),(2,-3,-1) and (3,-1,-2) are the vertices of an equilateral triangle

Show that points (2-sqrt3 , sqrt3 +1) , (1,0) and (3,2) form an equilateral triangle .

Prove that the triangle formed by joining the three points whose coordinates ere (1,2,3),(2,3,1) and (3,1,2) is an equilateral triangle.

Find the value of alpha , if the points (5,1), (-2,-3) and (8,2 alpha) are collinear.

Find the values of alpha so that the point P(alpha^(2),alpha) lies inside or on the triangle formed by the lines x-5y+6=0,x-3+2=0 and x-2y-3=0

If alpha_(1),alpha_(2),alpha_(3),beta_(1),beta_(2),beta_(3), are the values of n for which sum_(r=0)^(n-1)x^(2r), is divisible by sum_(r=0)^(n-1)x^(r) then prove that the triangle having vertices (alpha_(1),beta_(1)),(alpha_(2),beta_(2)) and (alpha_(3),beta_(3)) cannot be an equilateral triangle.

If the centroid of triangles formed by the vertices (1, 2, 3), (2, 1, 0) and (3, 1, 4) is (alpha, beta, gamma) then the value of [alpha]+[beta]+[gamma], where [ ] represents the greatest integer function, is _______.

Determine all the values of alpha for which the point (alpha,alpha^(2)) lies inside the triangle formed by the lines.2x+3y-1=0x+2y-3=05x-6y-1=0

Three point masses m_(1), m_(2) and m_(3) are located at the vertices of an equilateral triangle of side alpha . What is the moment of inertia of the system about an axis along the altitude of the triangle passing through m_(1)?