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A298860
Primitive cyclic quadrilaterals with integer area.
3
1, 3, 6, 8, 18, 12, 1, 5, 5, 7, 18, 16, 1, 2, 8, 9, 20, 12, 1, 5, 5, 9, 20, 15, 1, 4, 7, 8, 20, 18, 2, 5, 5, 8, 20, 20, 2, 5, 5, 10, 22, 18, 3, 5, 5, 9, 22, 24, 2, 4, 7, 11, 24, 20, 3, 5, 5, 11, 24, 21, 4, 5, 5, 10, 24, 28, 2, 6, 7, 9, 24, 30, 4, 5, 5, 12, 26, 24, 3, 4, 8, 11, 26, 30, 4, 5, 7, 10, 26, 36, 2, 5, 10, 11, 28, 36, 1, 7, 8, 14, 30, 28, 1, 8, 9, 12, 30, 42
OFFSET
1,2
COMMENTS
Entries are listed as sextuples: (a,b,c,d), Perimeter, Area. They are ordered first by perimeter, second by area, third by a, then b, then c, then d. Rectangles and kites with two right angles are not listed; thus a < b <= c <= d. By "primitive" we mean (a,b,c,d) is not a multiple of any earlier quadruple.
We observe that the number of odd integers in any quadruple is always an even number.
LINKS
Eric Weisstein's World of Mathematics, Cyclic Quadrilateral
EXAMPLE
The first row of the table gives sidelengths (a,b,c,d)=(1,3,6,8) with perimeter=18 and area=12. Thus:
a b c d Perim Area
= = = == ===== ====
1 3 6 8 18 12
1 5 5 7 18 16
1 2 8 9 20 12
1 5 5 9 20 15
1 4 7 8 20 18
2 5 5 8 20 20
2 5 5 10 22 18
3 5 5 9 22 24
2 4 7 11 24 20
3 5 5 11 24 21
4 5 5 10 24 28
etc.
KEYWORD
nonn,tabf
AUTHOR
STATUS
approved