This is not older than the rest, but old enough I believe: In 1775 Fagnano constructed periodic orbits for acutangular triangular billiards. The question about the existence of periodic oribits in general triangular (or poligonal) billiards (in the case of irrational angles) remains open. ( http://iml.univ-mrs.fr/editions/preprint2007/files/troubetzkoy_fagnano.pdf ).

This is not older than the rest, but old enough I believe: In 1775 Fagnano constructed periodic orbits for acutangular triangular billiards. The question about the existence of periodic orbits in general triangular (or polygonal) billiards (in the case of irrational angles) remains open. (

Troubetzkoy, Serge, Dual billiards, Fagnano orbits, and regular polygons, Am. Math. Mon. 116, No. 3, 251-260 (2009). arXiv:0704.0390, jstor. ZBL1229.37033, MR2491981. ).

This is not older than the rest, but old enough I believe: In 1775 Fagano constructed periodic orbits for acutangular triangular billiards. The question about the existence of periodic oribits in general triangular (or poligonal) billiards (in the case of irrational angles) remains open. ( http://iml.univ-mrs.fr/editions/preprint2007/files/troubetzkoy_fagnano.pdf ).

This is not older than the rest, but old enough I believe: In 1775 Fagnano constructed periodic orbits for acutangular triangular billiards. The question about the existence of periodic oribits in general triangular (or poligonal) billiards (in the case of irrational angles) remains open. ( http://iml.univ-mrs.fr/editions/preprint2007/files/troubetzkoy_fagnano.pdf ).