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Chapter 5
Problem 15 Saishi Shinzan is an unpublished manuscript edited by Nakamura Tokikazu, which contains a record of 208 sangaku dating from 1731 to 1828. One of the problems contained in Saishi Shinzan is this one, proposed in 1821 by Adachi Mitsuaki, and dedicated to the Asakusa Kanzeondo¯ temple, Tokyo. Advice: Do the previous problem first. Consider an infinite number of connected squares with sides ln (n = 1, 2, 3, . . .) in a right triangle (see figure 5.14). Let L be the total length of the sides, L = ∑n∞=1l n , and S be the area of the triangle minus the total area of the squares. Find l1 in terms of L and S. The solution is on page 175.
B S
l1
a
l2 l3
Figure 5.14. Find l1 in terms of L = Σln and S.
l4 C
A b
Problem 16 This problem can be seen as the fifth from the right top corner of the Katayamahiko shrine sangaku, color plate 5. A chain of circles of radii r 1, r 2 , r 3, and r4 is inscribed in the right triangle ABC, as shown in figure 5.15. Between the circles of radius r n are three smaller circles, t 1, t 2 and t 3, each of which touches two of the larger circles and is tangent to BC. Show that t 1t 3 = t 22 See page 176 for a solution.