Euler Piruleta Vs. Himself

The evil Lords of Matfor have created an Euler Piruleta Clone using a technique they learnt thanks to compulsively playing two thousand hours to Tomb Raider I.

Euler faces his fake-himself for the first time while he is visiting a round square. Both of them freak out and start running in opposite directions at the same speed.

Every time they meet, they freak out, and run changing the directions with one little difference: after every future encounter one of them will change his speed to half of his previous speed.

They change their speeds alternatively (let’s say that Euler is the first who changes his speed. Then after they meet, FakEuler changes the speed, after the next meeting, Euler changes his speed once again, and then FakEuler… this goes on forever and ever…)

How many times do they have to meet before they face themselves once again at the start position?

Deadline: Monday, 28th of April
Printable version available

Every correct answer will be awarded with 30 coins if only the answer is given and with 30 extra coins (making a total of 60) if a thorough explanation is featured.

Lara Croft Versus Doppleganger

PS: IBatman, V-Lo, Cesar Romero and The Headhunter have been awarded with 40 coins for solivng our last challenge!