Isoptic Point of the Non-Cyclic Quadrangle in the Isotropic Plane

Ema Jurkin, Marija Šimić Horvath () and Vladimir Volenec
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Ema Jurkin: Faculty of Mining, Geology and Petroleum Engineering, University of Zagreb, Pierottijeva 6, 10 000 Zagreb, Croatia
Marija Šimić Horvath: Faculty of Architecture, University of Zagreb, Kačićeva 26, 10 000 Zagreb, Croatia
Vladimir Volenec: Faculty of Science, University of Zagreb, Bijenička Cesta 30, 10 000 Zagreb, Croatia

Mathematics, 2024, vol. 12, issue 22, 1-10

Abstract: We study the non-cyclic quadrangle A B C D in the isotropic plane and its isoptic point. This is a continuation of the research carried out in a few previous papers. There, we put the non-cyclic quadrangle in the standard position, which enables us to prove its properties using a simple analytical method. In the standard position, the special hyperbola x y = 1 is circumscribed to the quadrangle. Hereby, we use the same method to obtain several results related to the isoptic point of the non-cyclic quadrangle. The isoptic point T is the inverse image of points A ′ , B ′ , C ′ , D ′ with respect to circumcircles of B C D , A C D , A B D , A C D , respectively, where A ′ , B ′ , C ′ , D ′ are isogonal points to vertices A , B , C , D with respect to triangles B C D , A C D , A B D , A C D . The circumircles are seen from T under the equal angles. Our analysis is motivated by the Euclidean results already published in the literature.

Keywords: isotropic plane; non-cyclic quadrangle; diagonal points; isogonality; isoptic point (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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