On Certain Polynomials Associated with the Triangle

Blundon, W. J.

doi:10.2307/2687913

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1973

2021

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“…Apparently, this result was not exploited until Blundon in two important papers [2], [3], on the structure of triangle inequalities, chose these variables as his canonical ones and showed that the B.I. 's (best possible homogeneous triangle inequalities) of the form…”

Section: Introductionmentioning

confidence: 99%

On best quadratic triangle inequalities

Frucht

Klamkin

1973

Geom Dedicata

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Section: Introductionmentioning

confidence: 99%

On best quadratic triangle inequalities

Frucht

Klamkin

1973

Geom Dedicata

View full text Add to dashboard Cite

show abstract

On Some Remarkable Points and Line Segments in Triangles

Mal'tsev¹,

Monastyreva²

2021

Известия АлтГУ

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Let ra, rb, rc be the radii, and OA, OB, OC the centers of tangent circles at the vertices to the circumcircle of a triangle ABC and to the opposite sides. In the paper [Andrica D., Marinescu D.S. New interpolation inequalities to Euler's R≥2 // Forum Geometricorum. 2017. Vol. 17], the authors proved that 4/R £ 1/ra + 1/rb +1/rc £2/r. In the paper [Isaev I., Maltsev Yu., Monastyreva A. On some relations in geometry of a triangle // Journal of Classical Geometry. 2018. Vol. 4], it is given the following generalization of these inequalities: 1/ra + 1/rb +1/rc=2/R+1/r. In that paper, we find the area of the triangle OAOBOC (see Theorem 1). We prove some relations for the numbers R-ra, R-rb, R-rc, where R is the circumradius of a triangle ABC. Namely, we find the expressions 1/R-ra+1/R-rb + 1/R-rc и a/R-ra+b/R-rb + c/R-rc by means by the parameters p, R and r (see Theorem 2). We estimate these values (see Theorem 3). Finally, using the results of paper [Maltsev Yu., Monastyreva A. On some relations for a triangle // International Journal of Geometry. 2019. Vol. 8 (1)] and representing the expression of (1-cos(αβ))(1-cos(β-γ))(1-cos(α-γ)) by means of p, R, r, we prove new proof of the fundamental triangle inequality (see Corollary 2).

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On Certain Polynomials Associated with the Triangle

Cited by 2 publications

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On best quadratic triangle inequalities

On best quadratic triangle inequalities

On Some Remarkable Points and Line Segments in Triangles

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