TDK-trikotniki

na študijskem programu Predmetni učitelj

Sabina Boršić (Author), Bojan Hvala (Mentor)

Abstract

Najprej narišemo Vectenovo konfiguracijo: nad stranicami poljubnega trikotnika ▫$ABC$▫ navzven načrtamo kvadrate ▫$AUVB$▫, ▫$BWXC$▫ in ▫$ACYZ$▫. Premice ▫$XY$▫, ▫$UZ$▫ in ▫$VW$▫ predstavljajo nosilke stranic trikotnika ▫$A'B'C'$▫, ki ga imenujemo trikotnik, ki se dotika kvadratov trikotnika ▫$ABC$▫ oz. TDK-trikotnik trikotnika ▫$ABC$▫. S ponovitvijo konstrukcije TDK-trikotnika na trikotniku ▫$A'B'C'$▫ dobimo t. i. drugi TDK-trikotnik ▫$A''B''C''$▫ trikotnika ▫$ABC$▫. V magistrskem delu obravnavamo podobnost osnovnega in prvega oz. drugega TDK-trikotnika. Trikotnik ▫$ABC$▫ in njegov TDK-trikotnik ▫$A'B'C'$▫ sta podobna, ko je osnovni trikotnik enakostraničen ali ko oglišče ▫$C$▫ leži na enem izmed dveh točno določenih krožnih lokov nad stranico ▫$AB$▫, pri čemer dve izmed možnosti predstavljata pravokotni trikotnik ▫$ABC$▫ s katetama v razmerju ▫$\sqrt{2}:1$▫. Trikotnik ▫$ABC$▫ in njegov drugi TDK-trikotnik ▫$A''B''C''$▫ sta vedno podobna. Izkaže se, da velja še več, in sicer da sta homotetična. Pristop k delu v glavnini temelji na podlagi elementarne geometrije, analitične geometrije ter na uporabi vektorjev in izometrij. Podana sta tudi dva dokaza, v katerih so za matematično sredstvo uporabljena kompleksna števila.

Keywords

magistrska dela;TDK-trikotnik;2TDK-trikotnik;podobnost;homotetičnost;kompleksna števila;Vectenova konfiguracija;

Data

Language:	Slovenian
Year of publishing:	2023
Typology:	2.09 - Master's Thesis
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	[S. Boršić]
UDC:	514.112.3(043.2)
COBISS:	157945603
Views:	197
Downloads:	8
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Square tangent triangles
Secondary abstract:	First, we draw Vecten's configuration: we draw squares ▫$AUVB$▫, ▫$BWXC$▫, and ▫$ACYZ$▫ outwards over the sides of an arbitrary triangle ▫$ABC$▫. The lines ▫$XY$▫, ▫$UZ$▫, and ▫$VW$▫ are joined and these joins are produced to form a triangle ▫$A'B'C'$▫, which we call a square tangent triangle of triangle ▫$ABC$▫. By repeating the construction of the square tangent triangle on triangle ▫$A'B'C'$▫, we obtain the so-called second square tangent triangle ▫$A''B''C''$▫ of triangle ▫$ABC$▫. In the master's thesis, we studied the similarity between the original triangle and the first or second square tangent triangle. Triangle ▫$ABC$▫ and its square tangent triangle ▫$A'B'C'$▫ are similar when the original triangle is equilateral or when vertex ▫$C$▫ lies on one of the two precisely determined circular arcs above the side ▫$AB$▫, where two of the possibilities represent a right triangle ▫$ABC$▫ with legs in the ratio ▫$\sqrt{2}:1$▫. Triangle ▫$ABC$▫ and its second square tangent triangle ▫$A''B''C''$▫ are always similar. Furthermore, it turns out that they are homothetic. Initially, we developed the entire story based on elementary geometry, analytical geometry, and the use of vectors and isometries. In the end, two proofs are also presented, in which we used complex numbers as a mathematical tool.
Secondary keywords:	master theses;square tangent triangle;second square tangent triangle;similarity;homotetic;complex numbers;Vecten configuration;Trikotnik;Kompleksna števila;Univerzitetna in visokošolska dela;
Type (COBISS):	Master's thesis/paper
Thesis comment:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages:	VIII, 45 f.
ID:	19256454