98.32 Triangles with two angles in the ratio 1:2

Nicollier, Grégoire

doi:10.1017/s0025557200008299

Cited by 2 publications

(2 citation statements)

References 2 publications

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“…Later Oppenheim, together with Daykin, explicitly characterized primitive integer triples for all p : q families, it seems fair to call them the Oppenheim triples. Their result was published back in 1967 [6], but special cases and related results were rediscovered later multiple times, see [2,3,7,11,13,15,16,17,21,24]. However, almost all attention went to the Oppenheim triples, while the "Pythagorean theorems", and algebro-geometric connections, were largely overlooked.…”

Section: Introductionmentioning

confidence: 99%

Algebraic geometry on imaginary triangles

Koshkin¹

2018

International Electronic Journal of Geometry

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We extend the notion of triangle to imaginary triangles with complex valued sides and angles, and parametrize families of such triangles by plane algebraic curves. We study in detail families of triangles with two commensurable angles, and apply the theory of plane Cremona transformations to find "Pythagorean theorems" for these triangles, which are interpreted as the implicit equations of their parametrizing curves.

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Section: Introductionmentioning

confidence: 99%

Algebraic geometry on imaginary triangles

Koshkin¹

2018

International Electronic Journal of Geometry

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“…Later Oppenheim, together with Daykin, explicitly characterized primitive integer triples for all p : q families, it seems fair to call them the Oppenheim triples. Their result was published in the Monthly back in 1967 [6], but special cases and related results were rediscovered later multiple times, see [2,3,7,11,13,15,16,17,21,24]. However, almost all attention went to the Oppenheim triples, while the "Pythagorean theorems", and algebrogeometric connections, were largely overlooked.…”

Section: Introductionmentioning

confidence: 99%

Imaginary triangles, Pythagorean theorems, and algebraic geometry

Koshkin

2017

Preprint

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We extend the notion of triangle to "imaginary triangles" with complex valued sides and angles, and parametrize families of such triangles by plane algebraic curves. We study in detail families of triangles with two commensurable angles, and apply the theory of plane Cremona transformations to find "Pythagorean theorems" for them, which are interpreted as the implicit equations of their parametrizing curves.

show abstract

98.32 Triangles with two angles in the ratio 1:2

Cited by 2 publications

References 2 publications

Algebraic geometry on imaginary triangles

Algebraic geometry on imaginary triangles

Imaginary triangles, Pythagorean theorems, and algebraic geometry

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