Existence of periodic orbits in rational billiards Recently I've got interested in dynamical billiards. Some results in this field are obtained by elementary methods. For instance, see George W. Tokarsky's  Polygonal Rooms Not Illuminable from Every Point or Andrew M. Baxter and Ron Umble's Periodic Orbits of Billiards on an Equilateral Triangle. Then I stumbled across this 

Every rational billiard has periodic orbits.

I tried to find a proof which was comprehensible to a freshman as I am, but I couldn't. Almost every article I had a look at somehow brought me to Howard Masur and Serge Tabachnikov's Rational billiards and ﬂat structures, which is mostly beyond my knowledge. My question is, is it possible to prove that theorem without Teichmuller spaces, quadratic diﬀerentials, ergodicity,...? If not, what background is needed to deal with this and related problems?
 A: 
Schwartz, Richard Evan. Mostly surfaces. Vol. 60. American Mathematical Society, 2011.

On p.219ff of Schwartz's book, he sketches "an elementary proof, due to Boshernitsyn, that every rational polygon has at least one periodic billiard path."
(He doesn't cite an explicit reference for Boshernitsyn's proof.)

                   


In answer to "what background is needed...?": The first 218 pages of Schwartz's
book. :-)
Less flippantly, you could essentially read backwards from the proof sketch
to the concepts that precede it.
A: You may go directly to the original result by Mazur himself 
H. Masur, Closed trajectories for quadratic differentials with an application to billiards, Duke Math. J. 53 (1986), 307-313
or a stronger theorem on the density in this Transactions AMS paper.
For a good read and background take a look at the book by Hurt, 
"Quantum Chaos and Mesoscopic Systems: Mathematical Methods in the Quantum".
A: The book of Tabachnikov
https://www.math.psu.edu/tabachni/Books/billiardsgeometry.pdf
is fine as an introducion to billiards
