I am reading this in my text:
Why do they say that $x = \frac{s}{2}$? In similar triangles, aren't the sides themselves supposed to be proportional? Why do they do this intermediate step using $\frac{s}{2}$?
Also, I forget, in an equilateral triangle, how do you know that the height bisects a side?
That is not what they are saying. They are saying that when using similar triangles the scale by which one side has increased is the same scale by which all sides increase. So $\frac{x}h$ and $\frac {s/2}{L/2}$ should be the same, since they represent the same scale factor.
The height is given by dropping a perpendicular down from one of the vertices of the triangle. Since we are looking at an equilateral triangle, each internal angle is $60^\circ$, so by dropping this perpendicular we have split the triangle into two triangles with internal angles $30^\circ,60^\circ,90^\circ$. Two of their sides are the same and all of their angles are the same, so the remaining side must also be equal. So we have cut the base in half.