TDK-trikotniki : na študijskem programu Predmetni učitelj

Boršić, Sabina

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Title:	TDK-trikotniki : na študijskem programu Predmetni učitelj
Authors:	ID Boršić, Sabina (Author) ID Hvala, Bojan (Mentor) More about this mentor...
Files:	EMAG_Borsic_Sabina_2023.pdf (1,05 MB) MD5: D51CEB75D2B45F1EED2A8CC6440B4027
Language:	Slovenian
Work type:	Master's thesis/paper
Typology:	2.09 - Master's Thesis
Organization:	FNM - Faculty of Natural Sciences and Mathematics
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:	TDK-trikotnik, 2TDK-trikotnik, podobnost, homotetičnost, kompleksna števila, Vectenova konfiguracija
Place of publishing:	Maribor
Place of performance:	Maribor
Publisher:	[S. Boršić]
Year of publishing:	2023
Number of pages:	VIII, 45 f.
PID:	20.500.12556/DKUM-84477
UDC:	514.112.3(043.2)
COBISS.SI-ID:	157945603
Publication date in DKUM:	06.07.2023
Views:	791
Downloads:	83
Metadata:
Categories:	FNM
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Licences

License:	CC BY-NC-ND 4.0, Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International

Link:	http://creativecommons.org/licenses/by-nc-nd/4.0/
Description:	The most restrictive Creative Commons license. This only allows people to download and share the work for no commercial gain and for no other purposes.
Licensing start date:	13.06.2023

Secondary language

Language:	English
Title:	Square tangent triangles : magistrsko delo
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.
Keywords:	square tangent triangle, second square tangent triangle, similarity, homotetic, complex numbers, Vecten configuration

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