Less elegant solutions to some problems

U radu su ponuđena rješenja zadataka i dokazi tvrdnji različitog karaktera. Iako je neke od postavljenih zadataka možda moguće riješiti, a neke tvrdnje dokazati, na elegantniji način, dana rješenja, odnosno dokazi, nam mogu poslužiti kao motivacija za proučavanje korištenih pojmova, a mogu nam poslužiti i kao ideje za rješavanje sličnih zadataka i dokazivanje drugih tvrdnji.The paper offers solutions to problems and proofs of claims of different nature. Although it might be possible to solve some of the tasks and prove some claims in a more elegant way, the given solutions and proofs can be used as a motivation for studying the concepts used, and also as ideas for solving similar tasks and proving other claims

On some properties of Kiepert parabola in the isotropic plane

n this paper we consider the curve which is an envelope of the axes of homology of a given triangle and the corresponding Kiepert triangles in the isotropic plane - the Kiepert parabola of the given triangle. We derive the equation of this parabola by using appropriate coordinate system. We give some new significant characterizations of this curve which are not valid in the Euclidean plane. We have also studied the relationships between Kiepert parabola and the Steiner point, the tangential triangle as well as the Jerabek hyperbola of the given triangle

Afino pravilan ikozaedar upisan u afino pravilan oktaedar u GS-kvazigrupi

A golden section quasigroup or shortly a GS-quasigroup is an idempotent quasigroup which satises the identities adot (ab dot c) dot c = b; adot (a dot bc) dot c = b. The concept of a GS-quasigroup was introduced by VOLENEC. A number of geometric concepts can be introduced in a general GS-quasigroup by means of the binary quasigroup operation. In this paper, it is proved that for any affine regular octahedron there is an affine regular icosahedron which is inscribed in the given affine regular octahedron. This is proved by means of the identities and relations which are valid in a general GS-quasigrup. The geometrical presentation in the GS-quasigroup C(frac{1}{2} (1 +sqrt{5})) suggests how a geometrical consequence may be derived from the statements proven in a purely algebraic manner.Kvazigrupa zlatnog reza ili kraće GS-kvazigrupa idempotentna je kvazigrupa u kojoj vrijede identiteti adot (ab dot c) dot c = b; adot (a dot bc) dot c = b. Pojam GS-kvazigrupe uveo je VOLENEC. Razni geometrijski pojmovi mogu biti uvedeni u GS-kvazigrupi pomoću binarne operacije te kvazigrupe. Korištenjem relacija i identiteta u općoj GS-kvazigrupi u ovom je radu pokazano da se svakom afino pravilnom oktaedru može upisati afino pravilan ikozaedar. Geometrijski prikaz u kvazigrupi C(frac{1}{2} (1 +sqrt{5})) pokazuje kako geometrijske tvrdnje mogu biti posljedica potpuno algebarskih razmatranja

Thebault circles of the triangle in an isotropic plane

In this paper the existence of three circles, which touch the circumscribed circle and Euler circle of an allowable triangle in an isotropic plane, is proved. Some relations between these three circles and elements of a triangle are investigated. Formulae for their radii are also given

Brocard angle of the standard triangle in an isotropic plane

The concept of Brocard angle of the standard triangle in an isotropic plane I2 is introduced. The relationships between Brocard angles of the allowable triangle and circum-Ceva’s triangle of its centroid and circum-Ceva’s triangle of its Feuerbach point are investigated

Cosymmedian triangles in isotropic plane

In this paper the concept of cosymmedian triangles in an isotropic plane is defined. A number of statements about some important properties of these triangles will be proved. Some analogies with the Euclidean case will also be considered

Affine Fullerene C 60

It will be shown that the affine fullerene C60, which is defined as an affine image of buckminsterfullerene C60, can be obtained only by means of the golden section. The concept of the affine fullerene C60 will be constructed in a general GS-quasigroup using the statements about the relationships between affine regular pentagons and affine regular hexagons. The geometrical interpretation of all discovered relations in a general GS-quasigroup will be given in the GS-quasigroup C(1/2(1+5))

Reciprocity in an isotropic plane

The concept of reciprocity with respect to a triangle is introduced in an isotropic plane. A number of statements about the prop erties of this mapping is proved. The images of some well known elements of a triangle with respect to this mapping will be studied