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

Two Steiner theorems about complete quadrilaterals

U radu proučavamo potpuni četverostran u euklidskoj ravnini. Četverostran ima niz zanimljivih svojstava. U radu dokazujemo dvije tvrdnje, koje je veliki geometar Jakob Steiner objavio 1827. godine, i to bez dokaza. U potpunom četverostranu se simetrale kutova sijeku u 16 točaka, koje su središta upisanih i pripisanih kružnica četiri trokuta. Steiner je ustvrdio da ovih 16 sjecišta leži četiri po četiri na ukupno osam kružnica i to svako od njih na po dvije od tih kružnica. Steiner je dalje utvrdio da tih osam kružnica tvore dvije četvorke kružnica, tako da je svaka kružnica iz jedne četvorke ortogonalna na svaku kružnicu iz druge četvorke. Tvrdnje su nakon objave mnogo puta dokazivane, a ovdje ćemo dati jedan njihov dokaz.In this paper, we study a complete quadrilateral in the Euclidean plane. The quadrilateral has a lot of interesting properties. Here we prove two claims, which were published by the great geometer Jakob Steiner in 1827, without proof. In a complete quadrilateral, the bisectors of angles are concurrent at 16 points, which are the incenters and excenters of the four triangles. Steiner asserted that these 16 intersections lie four by four on a total of eight circles, each of them on two of these circles. Steiner also found out that these eight circles form two quadruplets, so that each circle from one quadruplet is orthogonal to each circle from the other quadruplet. The claims have been proven many times since then, and here we give one of their proofs. The claims have since been proven many times, and here we will give one of their proofs

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

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

Vectors and transfers in hexagonal quasigroup

Hexagonal quasigroup is idempotent, medial and semisymmetric quasigroup. In this article we define and study vectors, sum of vectors and transfers. The main result is the theorem on isomorphism between the group of vectors, group of transfers and the Abelian group from the characterization theorem of the hexagonal quasigroups