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?