Let $ABC$ be an equilateral triangle with side length 2. Point $A'$ is chosen on side $BC$ such that the length of $A'B$ is $k < 1$. Likewise points $B'$ and $C'$ are chosen on sides $CA$ and $AB$ with $CB' = AC' = k$. Line segments are drawn from points $A', B', C'$ to their corresponding opposite vertices. The intersections of these line segments form a triangle, labeled $PQR$. Show that $PQR$ is an equilateral triangle with side length $\dfrac{4(1-k)}{\sqrt{k^{2}-2k+4}}$.