H. P. Young showed that there is a one-to-one correspondence between affine triple systems (or Hall triple systems) and exp. 3-Moufang loops (ML). Recently, L. Beneteau showed that (i) for any non-associative exp. 3- ML ( E, · ) with ‖ E‖ = 3 n , 3 ⩽ ‖ Z( E)‖ ⩽ 3 n−3 , where n ⩾ 4 and Z( E) is an associative center of ( E, ·), and (ii) there exists exactly one exp. 3- ML, denoted by ( E n , ·), such that ‖ E n ‖ = 3 n and ‖ Z( E n )‖ = 3 n−3 for any integer n ⩾ 4. The purpose of this paper is to investigate the geometric structure of the affine triple system derived from the exp. 3- ML( E n , ·) in detail and to compare with the structure of an affine geometry AG( n, 3). We shall obtain (a) a necessary and sufficient condition for three lines L 1, L 2 and L 3 in ( E n , ·) that the transitivity of the parallelism holds for given three lines L 1, L 2 and L 3 in ( E n , ·) such that L 1‖ L 2 and L 2‖ L 3 and (b) a necessary and sufficient condition for m + 1 points in E n (1 ⩽ m < n) so that the subsystem generated by those m + 1 points consists of 3 m points. Using the structure of hyperplanes in ( E n , ·), the p-rank of the incidence matrix of the affine triple system derived from the exp. 3- ML( E n , ·) is given.