Ceva's theorem when the point of concurrency lies on the side of the triangle $S$ tend towards Point $D$ along $AD$">
In the figure suppose the point $S$ tends to $D$ along Line $AD$ .
According to Ceva 
$$ \frac{AF}{FB} \cdot\frac{BD}{DC}\cdot\frac{CE}{EA} =1$$
as we see if S tends to D    $ \frac{AF}{FB} $ increases and $ \frac{CE}{EA} $ decreases 
And Ceva's theorem is valid and in the limiting case it seems to be valid, but is it valid when the point lies on the triangle, and what does the formula look like when the point is on the triangle?

It's not valid when $S=D$, because what that formula would then contain is a division by $0$, the length of $FB$. That's nonsensical. On the other hand, what would hold is
$$
AF\cdot BD\cdot CE=FB\cdot DC\cdot EA
$$
For any other case is exactly the same result, but it doesn't turn into garbage if $S$ lies on one of the sides of the triangle. 
This result is, however, a bit more difficult to remember, and a bit more difficult to make some kind of direct geometric sense of. I suspect that this is why the variation with ratios is more prevalent.
