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

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 $boldsymbol{v}_i=[x_{i},y_{i},z_{i}]$ $(i=1,2)$ have the scalar product $2Delta (alpha x_{1}x_{2}+beta y_{1}y_{2}+gamma z_{1}z_{2})$. 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)

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 $C( frac{1}{2}(1+sqrt 5 ))$ is given. The connection of GS-deltoids with parallelograms, GS-trapezoids, DGS-trapezoids and affine regular pentagons in the general GS-quasigroup is obtained