A348143 Areas of integer-sided cyclic quadrilaterals whose area equals their perimeter.

16, 18, 20, 30

Offset: 1

Author

Frank M Jackson, Oct 02 2021

Keywords

• fini
• full
• nonn

Comments

There are no further terms. Note that without the condition "integer-sided" there are other solutions, such as (1, 17/2, 17/2, 16) which has perimeter and area 34.

Examples

			The areas or perimeters 16, 18, 20, 30 pertain respectively to cyclic quadrilaterals with sides (4, 4, 4, 4), (3, 3, 6, 6), (2, 5, 5, 8), (5, 5, 6, 14).
		

Links

• Ralph H. Buchholz and James A. MacDougall, Cyclic Polygons with Rational Sides and Area, 2001. JNT 128 (2008) 17-48

Crossrefs

Cf. A098030, A290451. First four terms of A161874.

Programs

Mathematica

lst={}; Do[s=(a+b+c+d)/2; If[s>a, (K=Sqrt[(s-a)(s-b)(s-c)(s-d)]; If[IntegerQ[K]&&K==2s, AppendTo[lst, Sort@{a,b,c,d}]])], {a, 1, 15}, {b, 1, a}, {c, 1, b}, {d, 1, c}]; lst