{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:17:38Z","timestamp":1760145458464,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,7,17]],"date-time":"2024-07-17T00:00:00Z","timestamp":1721174400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The possible positions of an equilateral triangle whose vertices are located on the support sides of a generic triangle are studied. Using complex coordinates, we show that there are infinitely many such configurations, then we prove that the centroids of these equilateral triangles are collinear, defining two lines perpendicular to the Euler\u2019s line of the original triangle. Finally, we obtain the complex coordinates of the intersection points and study some particular cases.<\/jats:p>","DOI":"10.3390\/axioms13070478","type":"journal-article","created":{"date-parts":[[2024,7,17]],"date-time":"2024-07-17T15:15:19Z","timestamp":1721229319000},"page":"478","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Some Properties of the Equilateral Triangles with Vertices Located on the Support Sides of a Triangle"],"prefix":"10.3390","volume":"13","author":[{"given":"Dorin","family":"Andrica","sequence":"first","affiliation":[{"name":"Faculty of Mathematics and Computer Science, Babe\u015f-Bolyai University, 400084 Cluj-Napoca, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4193-9842","authenticated-orcid":false,"given":"Ovidiu","family":"Bagdasar","sequence":"additional","affiliation":[{"name":"School of Computing, University of Derby, Derby DE22 1GB, UK"},{"name":"Department of Mathematics, Faculty of Exact Sciences, \u201c1 Decembrie 1918\u201d University of Alba Iulia, 510009 Alba Iulia, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,17]]},"reference":[{"key":"ref_1","unstructured":"Apr\u00f3, J. (2023). Triangles and Quadrilaterals Inscribed in Jordan Curves. [Ph.D. Thesis, E\u00f6tv\u00f6s Lor\u00e1nd University]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4064\/fm-110-1-1-9","article-title":"Equilateral triangles and continuous curves","volume":"110","author":"Meyerson","year":"1980","journal-title":"Fund. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1007\/BF00151519","article-title":"Triangles inscribed in simple closed curves","volume":"43","author":"Nielsen","year":"1992","journal-title":"Geom. Dedicata"},{"key":"ref_4","unstructured":"Gupta, A., and Rubinstein-Salzedo, S. (2021). Inscribed triangles of Jordan curves in Rn. arXiv."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/bull\/1755","article-title":"Rectangles, curves, and Klein bottles","volume":"59","author":"Schwartz","year":"2022","journal-title":"Bull. Amer. Math. 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Geometric Transformations II, Random House. Translated from Russian.","DOI":"10.5948\/UPO9780883859360"},{"key":"ref_12","first-page":"113","article-title":"Inscribed equilateral triangles in general triangles","volume":"13","author":"Andrica","year":"2024","journal-title":"Int. J. Geom."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1007\/s00022-007-1751-z","article-title":"Configurations of inscribed equilateral triangles","volume":"87","year":"2007","journal-title":"J Geom."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Andreescu, T., and Andrica, D. (2014). Complex Numbers from A to \u2026Z, Birkh\u00e4user. [2nd ed.].","DOI":"10.1007\/978-0-8176-8415-0"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/7\/478\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:18:32Z","timestamp":1760109512000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/7\/478"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,17]]},"references-count":14,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2024,7]]}},"alternative-id":["axioms13070478"],"URL":"https:\/\/doi.org\/10.3390\/axioms13070478","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,7,17]]}}}