46 research outputs found
αβγσ - technology in the triangle geometry
The barycentric coordinates of the most important points and circles and the equations of the most important lines, conics and cubics of the geometry of triangle ABC are expressed by means of numbers
α = cotA, β = cotB, γ = cotC, σ = cotC and σ = α+β+γ
A generalization of the butterfly theorem
In this paper a new generalization of the well-known butterfly theorem is given using the complex coordinates
Circles in barycentric coordinates
Let ABC be a fundamental triangle with the area ∆. For a circle K the powers of vertices A,B,C with regard to K divided by 2∆ are said to be the barycentric coordinates of K with respect to triangle ABC. This paper gives some theory and applications of these coordinates
Metrical relations in barycentric coordinates
Let Δ be the area of the fundamental triangle ABC of barycentric coordinates and α=cot A,β =cot B, γ=cot C. The vectors have the scalar product . This fact implies all important formulas about metrical relations of points and lines. The main and probably new results are Theorems 1 and 8
A note on medial quasigroups
In this short note we prove two results about medial quasigroups.
First, let φ and ψ be binary operations defined by multiplication, left and right division in a medial quasigroup. Then φ and ψ are mutually medial, i.e. φ(ψ(a,b),ψ(c,d))=ψ(φ(a,b),φ(c,d)).
Second, four points a, b, c, d in an idempotent medial quasigroup form a parallelogram if and only if d=(a/b)(bc)
A note on medial quasigroups
In this short note we prove two results about medial quasigroups.
First, let φ and ψ be binary operations defined by multiplication, left and right division in a medial quasigroup. Then φ and ψ are mutually medial, i.e. φ(ψ(a,b),ψ(c,d))=ψ(φ(a,b),φ(c,d)).
Second, four points a, b, c, d in an idempotent medial quasigroup form a parallelogram if and only if d=(a/b)(bc)
GS-trapezoids in GS-quasigroups
In this paper the concept of a GS-trapezoid in a GS-quasigroup is defined and some characterizations of that are proved and geometrical representation of the properties of the quaternary relation GST in the GS-quasigroup is given
Stammler\u27s circles, Stammler\u27s triangle and Morley\u27s triangle of a given triangle
By means of complex coordinates shorter proofs of the results of L. Stammler [1], [2] will be given, plus several statements connected with them
GS-deltoids in GS-quasigroups
A "geometric\u27\u27 concept of the GS-deltoid is introduced and investigated in the general GS-quasigroup and geometrical interpretation in the GS-quasigroup is given. The connection of GS-deltoids with parallelograms, GS-trapezoids, DGS-trapezoids and affine regular pentagons in the general GS-quasigroup is obtained