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On Brocard Points of Harmonic Quadrangle in I_2(R)


Šimić Horvath, Marija; Jurkin, Ema; Volenec, Vladimir; Beban-Brkić, Jelena
On Brocard Points of Harmonic Quadrangle in I_2(R) // Abstracts − 19th Scientific-Professional Colloquium on Geometry and Graphics, Starigrad- Paklenica, September 4 – 8, 2016 / Došlić, Tomislav ; Jurkin, Ema (ur.).
Zagreb, 2016. str. 54-55 (predavanje, nije recenziran, sažetak, ostalo)


Naslov
On Brocard Points of Harmonic Quadrangle in I_2(R)

Autori
Šimić Horvath, Marija ; Jurkin, Ema ; Volenec, Vladimir ; Beban-Brkić, Jelena

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, ostalo

Izvornik
Abstracts − 19th Scientific-Professional Colloquium on Geometry and Graphics, Starigrad- Paklenica, September 4 – 8, 2016 / Došlić, Tomislav ; Jurkin, Ema - Zagreb, 2016, 54-55

Skup
19th Scientific-Professional Colloquium on Geometry and Graphics

Mjesto i datum
Starigrad-Paklenica, Hrvatska, 04-08.09.2016

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Nije recenziran

Ključne riječi
Isotropic plane ; harmonic quadrangle ; Brocard points

Sažetak
In this talk we present several results concerning the geometry of a harmonic quadrangle in the isotropic plane I_2(R). We consider the standard cyclic quadrangle with the circumscribed circle given by y = x^2 and the vertices chosen to be A = (a, a^2), B = (b, b^2), C = (c, c^2), and D = (d, d^2), with a, b, c, d being mutually different real numbers, a < b < c < d. The harmonic quadrangle in the isotropic plane is a standard cyclic quadrangle with a special property: vertices A, B, C, and D are chosen in a way that tangents A and C at the vertices A and C, respectively, intersect in the point incident with BD, and tangents B and D at the vertices B and D, respectively, are intersected in the point incident with AC. We show that there exist a unique point P_1, so-called the first Brocard point, such that the lines P_1A, P_1B, P_1C, and P_1D form equal angles with the sides AB, BC, CD and DA, respectively. Similarly, the second Brocard point is defined as the point such that the lines P_2A, P_2B, P_2C, and P_2D form equal angles with the sides AD, DC, CB, and BA, respectively. We compare the obtained results with their Euclidean counterparts.

Izvorni jezik
Engleski

Znanstvena područja
Matematika



POVEZANOST RADA


Ustanove
Geodetski fakultet, Zagreb,
Prirodoslovno-matematički fakultet, Matematički odjel, Zagreb,
Arhitektonski fakultet, Zagreb,
Rudarsko-geološko-naftni fakultet, Zagreb