Question 17
Points $A'$, $B'$ and $C'$ are chosen correspondingly on the sides $AB$, $BC$, and $CA$ of an equilateral triangle $ABC$ so that $\frac{|A'B|}{|AB|} = \frac{|B'C|}{|BC|} = \frac{|C'A|}{|CA|} = k$. Find all positive real numbers $k$ for which the area of the triangle $A'B'C'$ is exactly half of the area of the triangle $ABC$.