A298860 Primitive cyclic quadrilaterals with integer 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, 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

Author

Gregory Gerard Wojnar, Jan 27 2018

Keywords

• nonn
• tabf

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.

Examples

			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.
		

Links

• Eric Weisstein's World of Mathematics, Cyclic Quadrilateral
• Wikipedia, Cyclic quadrilateral

Crossrefs

Cf. A298907, A297790, A210250, A230136, A131020, A218431, A219225, A233315, A242778, A273691, A273890.