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不久前,学校进行了一次考试,普遍认为最后一题过繁。原题如下: 设扇形AOB的半径为a,中心角为θ(锐角)。由A向半径OB引垂线AB_1,由垂足B_1引弦AB的平行钱交OA于A_1.再由A_l引半径OB的垂线AB_2,再由B_2引弦AB的平行线交OA于A_2,这样无限地反复地继续作下去。所得△ABB_1,△A_1B_1B_2,△A_nB_nB_(n+L)…的面积分别为S_1,S_2,S_3,…,s~n,…求所有这些三角形面积的和。
Not long ago, the school conducted an exam and it was generally believed that the last question was too complicated. The original question is as follows: Let the radius of the fan-shaped AOB be a, and the center angle be θ (acute). From A to the radius OB leads to the vertical line AB_1, and the parallel money of the AB footing AB approaches OA to A_1. Then the vertical line AB_2 of the radius OB is drawn by the A_1, and then the parallel line AB of the B_2 pickup string AB crosses the OA to A_2. This continues indefinitely. The areas of ΔABB_1, ΔA_1B_1B_2, ΔA_nB_nB_(n+L)... are respectively S_1, S_2, S_3,..., s~n,... Find the sum of all these triangle areas.