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
