An equilateral triangle ABC with side length 1 unit has points D and E marked on sides AB and AC respectively, where these sides are divided into three equal parts. Let K be the midpoint of the line segment DE. A new equilateral triangle is drawn with one vertex at K and the opposite side on BC, and the same process is applied to the newly drawn equilateral triangles. What is the sum of the areas of all nested triangular regions drawn in this manner, in square units? A) $\frac { \sqrt { 3 } } { 3 }$ B) $\frac { 3 \sqrt { 3 } } { 4 }$ C) $\frac { 8 \sqrt { 3 } } { 9 }$ D) $\frac { 5 \sqrt { 3 } } { 16 }$ E) $\frac { 9 \sqrt { 3 } } { 32 }$
An equilateral triangle ABC with side length 1 unit has points D and E marked on sides AB and AC respectively, where these sides are divided into three equal parts. Let K be the midpoint of the line segment DE. A new equilateral triangle is drawn with one vertex at K and the opposite side on BC, and the same process is applied to the newly drawn equilateral triangles.
What is the sum of the areas of all nested triangular regions drawn in this manner, in square units?\\
A) $\frac { \sqrt { 3 } } { 3 }$\\
B) $\frac { 3 \sqrt { 3 } } { 4 }$\\
C) $\frac { 8 \sqrt { 3 } } { 9 }$\\
D) $\frac { 5 \sqrt { 3 } } { 16 }$\\
E) $\frac { 9 \sqrt { 3 } } { 32 }$