Problem:

Through a point in the interior of a triangle $ABC$, three lines are drawn, one parallel to each side. These lines divide the sides of the triangle into three regions each. Let $a$, $b$, and $c$ be the lengths of the sides opposite $\angle A$, $\angle B$, and $\angle C$, respectively, and let $a'$, $b'$, and $c'$ be the lengths of the middle regions of the sides opposite $\angle A$, $\angle B$, and $\angle C$, respectively. Find the numerical value of $\dfrac{a'}{a} + \dfrac{b'}{b} + \dfrac{c'}{c}$.