The Brahmagupta Triangles

Beauregard, Raymond A.; Suryanarayan, E. R.

doi:10.1080/07468342.1998.11973907

Cited by 3 publications

(5 citation statements)

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“…Indeed, it was shown there how to find all Heron triangles with sides whose lengths form an arithmetic progression. At the same time Beauregard and Suryanarayan published two papers [1] and [2] in which they drew the same conclusion. In this paper, we extend the problem to search for triangles with rational area which have rational sides in geometric progression.…”

Section: Introductionmentioning

confidence: 68%

“…There are 8 points easily discovered by inspection: (0, 0), (1,0), (-3,0) (the obvious points of order 2), D and (3, ±6), (-1, ±2) which a calculation shows to be points of order 4. Since the discriminant A = 2 8 3 2 we have good reduction modulo 5; we find that |i? (F 5 )| = 8 and so there are no more points of finite order.…”

Section: Cyclic Quadrilaterals With Sides In Arithmetic Progressionmentioning

confidence: 94%

“…For completeness, we mention briefly the case of triangles with rational area having rational sides in arithmetic progression (as described in [5] and [2]). Let rational numbers a, b, and c be the sides of a triangle having rational area.…”

Section: Triangles With Sides In Arithmetic Progressionmentioning

confidence: 99%

“…In this section we consider the case of triangles with rational area having sides in geometric progression. If we let the sides be a, ar, ar 2 where a, r € Q and r ^ 0 then the semiperimeter is s = a ( l + r + r 2 )/2. As before we use Heron's formula to compute the area; this yields…”

Section: T R I a N G L E S W I T H S I D E S In G E O M E T R I C P Rmentioning

confidence: 99%

“…Dividing through by (d/2) 4 and setting W = AA/d 2 and to solve our problem we must find its group E(Q) of rational points.…”

Section: Cyclic Quadrilaterals With Sides In Arithmetic Progressionmentioning

confidence: 99%

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Heron quadrilaterals with sides in arithmetic or geometric progression

Buchholz¹,

MacDougall

1999

Bull. Austral. Math. Soc.

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We study triangles and cyclic quadrilaterals which have rational area and whose sides form geometric or arithmetic progressions. A complete characterisation is given for the infinite family of triangles with sides in arithmetic progression. We show that there are no triangles with sides in geometric progression. We also show that apart from the square there are no cyclic quadrilaterals whose sides form either a geometric or an axiuiiiitJLii; piugiesaiuii. ziie SUIULIUIL ui UULII 14ucu11ua.Le1.a1 ttisca mvuivca seaiuiiiiig for rational points on certain elliptic curves.

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Section: Introductionmentioning

confidence: 68%

Section: Cyclic Quadrilaterals With Sides In Arithmetic Progressionmentioning

confidence: 94%

Section: Triangles With Sides In Arithmetic Progressionmentioning

confidence: 99%

Section: T R I a N G L E S W I T H S I D E S In G E O M E T R I C P Rmentioning

confidence: 99%

“…Dividing through by (d/2) 4 and setting W = AA/d 2 and to solve our problem we must find its group E(Q) of rational points.…”

Section: Cyclic Quadrilaterals With Sides In Arithmetic Progressionmentioning

confidence: 99%