Triangle centres: some questions in Euclidean geometry

Hajja, Mowaffaq

doi:10.1080/00207390118743

Cited by 7 publications

(3 citation statements)

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“…78-79]. This still holds for many other pairs of centers as seen in [19], and fails for other rather artificial pairs as seen in [1]. Analogues for higher dimensional simplices are explored in [14,15,22,23] and [20], where the degree of regularity implied by the coincidence of two or more centers is investigated.…”

Section: Introductionmentioning

confidence: 99%

Coincidences of Centers of Plane Quadrilaterals

2009

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We consider the three natural centers of mass associated with a general quadrilateral together with the Fermat-Torricelli center and explore the degree of regularity implied by the coincidence of two or more of these four centers. We then extend this investigation to include cyclic quadrilaterals, where we add the circumcenter to the list, and circumscriptible quadrilaterals, where we add the incenter.

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Section: Introductionmentioning

confidence: 99%

Coincidences of Centers of Plane Quadrilaterals

2009

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“…We now add 3 other characterisations. The first appears as Theorem 5 (p. 32) in [9], and it states that…”

Section: References and Three More Characterisationsmentioning

confidence: 99%

Triangles whose sides form an arithmetic progression

Abu-Saymeh¹,

Hajja²

2020

Math. Gaz.

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This article is motivated by, and is meant as a supplement to, the recent paper [1]. That paper proves three geometric characterisations of triangles whose sides are in arithmetic progression, or equivalently triangles in which one of the sides is the arithmetic mean of the other two. More precisely, it gives three geometric contexts in which such triangles appear. In this Article, we supply references for the results in [1] and we provide more proofs of these results. We also add more contexts in which such triangles appear, and we raise related issues for future work. We hope that this will be a source of problems for training for, and for including in, mathematical competitions.

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“…Therefore Remark 9. It is well-known that if any two of the centers G, I, O, H of a triangle coincide, then the triangle is equilateral; see [2] and the references therein. It also goes without saying that all centers of an equilateral triangle coincide.…”

Section: Center Trilinears Barycentricsmentioning

confidence: 99%