the Cartesian co-ordinates of A, , A2 and A3 being (xt , yj, (x2 , y2) and (x3 , y3) respectively, intersect at L. Let the lines H3M and H,M be the lines derived from the lines H3L and H^L by the procedure by which in Fig. 2 the line 12 was derived from the line 34. Let them intersect at the point M, ysr). Using basic methods of analytical geometry we obtain for ?/."and a similar formula for xM • These formulae are symmetrical in the co-ordinates of A i , A 2 and A3 , so that the line IL.M derived from the line H2L by the same procedure passes through M. In other words, if the points 0, 0' and 0" in Fig. 2 are the mid-points of the sides of the triangle ABC, the line 02 for the side AB and the corresponding lines 0'2 and 0"2 for the sides BC and CA intersect in one point 2.If (see Fig. 3) the ratiothat corresponds to the ratio
Abstract.In this paper we derive the power series expansions of four infinite products of the formwhere the index sets Si and S2 are specified with respect to a modulus (Theorems 1, 3, and 4). We also establish a useful formula for expanding the product of two Jacobi triple products (Theorem 2). Finally, we give nonexistence results for identities of two forms.
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