Yang Hui Magic Circle [Part 1]

 
The sum of the numbers on four diameters
 
28 + 5 + 11 + 25 + 9 + 7 + 19 + 31 + 12 = 147 = (69 * 2) + 9
20 + 16 + 23 + 10 + 9 + 2 + 29 + 32 + 6 = 147
8 + 17 + 26 + 18 + 9 + 22 + 13 + 1 + 33 = 147
27 + 15 + 3 + 24 + 9 + 30 + 14 + 21 + 4 = 147
 
The sum of eight radii without 9
 
28 + 5 + 11 + 25 = 69
7 + 19 + 31 + 12 = 69
20 + 16 + 23 + 10 = 69
2 + 29 + 32 + 6 = 69
8 + 17 + 26 + 18 = 69
22 + 13 + 1 + 33 = 69
27 + 15 + 3 + 24 = 69
30 + 14 + 21 + 4 = 69

The sum of all numbers on each circle (not including 9)
 
10 + 22 + 7 + 30 + 2 + 18 + 25 + 24 = 138 = 2 * 69
23 + 13 + 19 + 14 + 29 + 26 + 11 + 3 = 138 = 2 * 69
16 + 1 + 31 + 21 + 32 + 17 + 5 + 15 = 138 = 2 * 69
20 + 33 + 12 + 4 + 6 + 8 + 28 + 27 = 138 = 2 * 69
 
The sum of the numbers on the semi-circles:
 
28 + 27 + 20 + 33 + 12 = 120
5 + 15 + 16 + 1 + 31 = 68
11 + 3 + 23 + 13 + 19 = 69
25 + 24 + 10 + 22 + 7 = 88

120 + 68 + 69 + 88 = 345

27 + 20 + 33 + 12 + 4 = 96
15 + 16 + 1 + 31 + 21 = 84
3 + 23 + 13 + 19 + 14 = 72
24 + 10 + 22 + 7 + 30 = 93

96 + 84 + 72 + 93 = 345

20 + 33 + 12 + 4 + 6 = 75
16 + 1 + 31 + 21 + 32 = 101
23 + 13 + 19 + 14 + 29 = 98
10 + 22 + 7 + 30 + 2 = 71

75 + 101 + 98 + 71 = 345

33 + 12 + 4 + 6 + 8 = 63
1 + 31 + 21 + 32 + 17 = 102
13 + 19 + 14 + 29 + 26 = 101
22 + 7 + 30 + 2 + 18 = 79

63 + 102 + 101 + 79 = 345

12 + 4 + 6 + 8 + 28 = 58
31 + 21 + 32 + 17 + 5 = 106
19 + 14 + 29 + 26 + 11 = 99
7 + 30 + 2 + 18 + 25 = 82

58 + 106 + 99 + 82 = 345

4 + 6 + 8 + 28 + 27 = 73
21 + 32 + 17 + 5 + 15 = 90
14 + 29 + 26 + 11 + 3 = 83
30 + 2 + 18 + 25 + 24 = 99

73 + 90 + 83 + 99 = 345

6 + 8 + 28 + 27 + 20 = 89
32 + 17 + 5 + 15 + 16 = 85
29 + 26 + 11 + 3 + 23 = 92
2 + 18 + 25 + 24 + 10 = 79

89 + 85 + 92 + 79 = 345

8 + 28 + 27 + 20 + 33 = 116
17 + 5 + 15 + 16 + 1 = 54
26 + 11 + 3 + 23 + 13 = 76
18 + 25 + 24 + 10 + 22 = 99
 
116 + 54 + 76 + 99 = 345
 
345 = 3 * 5 * 23   or   5 * 69