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Sufficient condition for 4 complex numbers to represent a rectangle

Source: Romanian District Olympiad 2011, Grade X, Problem 2

October 8, 2018
complex numbersabsolute valuegeometryrectangletrigonometry

Problem Statement

a) Show that if four distinct complex numbers have the same absolute value and their sum vanishes, then they represent a rectangle.
b) Let x,y,z,t x,y,z,t be four real numbers, and k k be an integer. Prove the following implication: ∑j∈{x,y,z,t}sin⁡j=0=∑j∈{x,y,z,t}cos⁡j  ⟹  ∑j∈{x,y,z,t}sin⁡(1+2n)j. \sum_{j\in\{ x,y,z,t\}} \sin j = 0 = \sum_{j\in\{ x,y,z,t\}} \cos j\implies \sum_{j\in\{ x,y,z,t\}} \sin (1+2n)j.
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