#16)What are the loci of points z which satisfy the following relations...?
d.) 0 < Re(iz) < 1 ???
g.) α < arg(z) < β, γ < Re(z) < δ, where -π/2 < αα, β < π/2, γ > 0 ???
I'm also wondering for help with this proof:
z_1 + z_2 + z_3 = 0 and |z_1| + |z_2| + |z_3| = 1,
prove this defines an equilateral triangle inscribed in the unit circle |z| = 1. Any hints? What should I show to prove this?
(PS - the book is available on Google Books for those interested in seeing the original problems. They're on page 9, numbers 15/16. Book is by Silverman and called Introductory Complex Analysis.)
The Attempt at a Solution
For #15, I thought I might be able to use the inequality relations for triangles of a + b < c, or two sides are always less than the length of the third side, but that got me nowhere.
Then I thought I might be able to use arg(z_1) + arg(z_2) + arg(z_3) = pi, but I can't figure out what to do with that or how to prove it.
For #16, I just need clarification. Does 'loci of pts z which satisfy....' mean the shape that this set of points inscribes?
I just want to make sure I understand the problem. Even if I am right, though, I'm not sure how to approach these! I know that's not much of an attempt but even a hint or guideline would be helpful. z can be so many things...