There are three snails each at a vertex of an equilateral triangle "The first sets out towards the second, the second towards the third and the third towards the first, with a uniform speed of 5 cm min −1. During their motion each of them always heads towards its respective target snail." - This problem is from the book "200 puzzling problems"
Now I do not understand the given solution for the trajectory, mainly how does the equation follow from the reasoning. I would be very grateful for a little suggestion here. 
These are obviously point snails rather than snails with a non-zero length, so it's best that this question has been migrated.
Consider n point snails arranged in a regular convex n-gon such that the point snails move in a coordinated manner (i.e., the same angular velocity) and such that at any point in time, each point snail moves in the direction of the neighbor directly in front of it with a constant speed v. Symmetry dictates that the n snails will maintain that regular convex n-gon shape until they simultaneously meet at the center of the n-gon.
The velocity vector of each of the n snails is $$\begin{aligned} \vec v &= v\left(-\cos\left(\frac{\pi}2 - \frac{\pi}n\right)\,\hat r + \sin\left(\frac{\pi}2 - \frac{\pi}n\right)\,\hat\theta\right) \\ &= v\left(-\sin\left(\frac{\pi}n\right)\,\hat r + \cos\left(\frac{\pi}n\right)\,\hat\theta\right) \end{aligned}$$ Equating this with the time derivative of $r\,\hat r = \dot r\,\hat r + r\dot \theta\,\hat\theta$ results in $$\begin{aligned} \dot r &= -v\sin\left(\frac{\pi}n\right) && (1)\\ r\dot\theta &= \phantom{-}v\cos\left(\frac{\pi}n\right) && (2) \end{aligned}$$
Equation (1) tells us that the radial distance decreases linearly with respect to time, which in turn tells us snails will meet at the center in finite time. Equation (2) tells us that the snails will be extremely dizzy when they meet. Their angular velocity diverges to infinity at the time they meet.
Re-expressing $r(t)$ as a function of angle $\theta$, equations (1) and (2) tell us the derivative of $r$ as a function of angle: $$\frac{\mathrm d r(\theta)}{\mathrm d \theta} = \frac{\dot r}{\dot\theta} = -r \tan\left(\frac{\pi}n\right) $$ The solution to this is the logarithmic spiral $$r(\theta) = r_0\exp\left(-\theta \tan\left(\frac{\pi}n\right) \right)$$
There's not much to see in the case of three snails. The triangle will have shrunk to about 7% of its original linear size after just $1/4$ of a rotation (90 degrees) and less than $1/2$% of its original linear size after $1/2$ of a rotation. The infinite number of rotations that follow will be pretty much invisible.