A triangle has corners at points A, B, and C. Side AB has a length of #7 #. The distance between the intersection of point A's angle bisector with side BC and point B is #6 #. If side AC has a length of #14 #, what is the length of side BC?

Given AB = 7, AC = 14 and BD = 6.

Using the Angle Bisector Theorem, the length of BC can be calculated as $BC = \frac{AB \times AC}{AB + AC}$. Substituting the given values, we get $BC = \frac{7 \times 14}{7 + 14}$, which simplifies to $BC = \frac{98}{21}$, or approximately $BC = 4.67$. Therefore, the length of side BC is approximately 4.67.

Using the angle bisector theorem, we can determine the length of side BC. According to the theorem, in a triangle, an angle bisector divides the opposite side into segments that are proportional to the lengths of the adjacent sides.

Let the point where the angle bisector of angle A intersects side BC be point D. Since AD is the angle bisector of angle A, we can set up the following proportion:

$\frac{BD}{DC} = \frac{AB}{AC}$

Plugging in the values we have:

$\frac{BD}{DC} = \frac{7}{14}$

$\frac{BD}{DC} = \frac{1}{2}$

Now, we know that BD = 6 (the distance between the intersection of the angle bisector with side BC and point B). We can set up another proportion using this information:

$\frac{BD}{DC} = \frac{6}{x}$

Where x is the length of side BC. Plugging in the values we have:

$\frac{6}{x} = \frac{1}{2}$

Solving for x:

$x = \frac{6}{\frac{1}{2}}$

$x = 6 \times 2$

$x = 12$

Therefore, the length of side BC is 12.