First, we will get how much time all three cyclists take to cover a circumference of 360 km 

So, the time taken for cyclist1 = 360/48 = 7.5 days

So, the time taken for cyclist2 = 360/60 = 6 days

So, the time taken for cyclist3 = 360/72  = 5 days

Here, LCM of 7.5, 6, 5 = 30

So, they will meet after 30 days.

Now, in 30 days, first cyclist will travel = 30/7.5 = 4 rounds

In 30 days, second cyclist will travel = 30/6 = 5 rounds

In 30 days, third cyclist will travel = 30/5 = 6 rounds

So, after 30 days, all the 3 cyclists will meet at starting point.