A generalization of the butterfly theorem

In this paper a new generalization of the well-known butterfly theorem is given using the complex coordinates

The butterfly theorem for conics

The butterfly theorem holds for any diameter of any conic

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

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 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)

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

Pascal-Brianschonovi skupovi u Pappusovim projektivnim ravninama

It is well-known that Pascal and Brianchon theorems characterize conics in a Pappian projective plane. But, using these theorems and their modifications we shall show that the notion of a conic (or better a Pascal-Brianchon set) can be defined without any use of theory of projectivities or of polarities as usually.Poznato je da Pascalov i Brianchonov teorem karakteriziraju kivulje 2. reda u Pappusovoj projektivnoj ravnini. Međutim, koristeći te teoreme i njihove modifikacije pokazat ćemo da se pojam krivulje 2. reda (ili bolje: pojam Pascal-Brianchonovog skupa) može definirati bez pomoći projektiviteta ili teorije polariteta, kao što se to obično radi

DGS-trapezoids in GS-quasigroups

The concept of the DGS-trapezoid is defined and investigated in any GS-quasigroup and geometrical interpretation in the GS-quasigroup $C( frac{1}{2}(1+sqrt 5 ))$ is also given. The connection of this concept with GS-trapezoids in the general GS-quasigroup is obtained

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