Question

If a(z1),b(z2),c(z3) are the vertices of an equilateral triangle abc , then value of arg((z2+z3-2z1)/(z3-z2)) is equal to

Answer 1

it is based on complex numbers. for better understanding assume a , b and c in such a way that it forms an equilateral triangle.

Let a(z1) = (-1, 0)
b(z2) = (1, 0)
c (z3) = (0, √3)
you can see , |z1 - z2| = |z2 - z3| = |z3 - z1| = 2

now, (z2 + z3 - 2z1) = 1 + i0 + 0 + i√3 - 2(-1 + i0)
= 1 + i√3 + 2 = 3 + i√3

(z3 - z2) = i√3 - 1 = -1 + i√3

so, (z2 + z3 - 2z2)/(z3 - z2) = (3 + i√3)/(-1 + i√3)
= (3 + i√3)(-1 - i√3)/{(-1)² - (i√3)²}
= (3 + i√3)(-1 - i√3)/(1 + 3)
= (-3 - i3√3 - i√3 + 3)/4
= -i√3
= 0 + i(-√3)

so, arg{(z2 + z3 - 2z1)/(z3 - z1)} = tan^-1(-√3/0)
= tan^-1(∞) = π/2

hence, answer is π/2