{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:21:28Z","timestamp":1760145688673,"version":"build-2065373602"},"reference-count":9,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,8,22]],"date-time":"2024-08-22T00:00:00Z","timestamp":1724284800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science and Technology Council in Taiwan","award":["NSTC-112-2221-E-013-003"],"award-info":[{"award-number":["NSTC-112-2221-E-013-003"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Malfatti\u2019s problem involves three circles (called Malfatti circles) that are tangent to each other and two sides of a triangle. In this study, our objective is to extend the problem to find 6, 10, \u2026 \u22111ni (n &gt; 2) circles inside the triangle so that the three corner circles are tangent to two sides of the triangle, the boundary circles are tangent to one side of the triangle, and four other circles (at least two of them being boundary or corner circles) and the inner circles are tangent to six other circles. We call this problem the extended general Malfatti\u2019s problem, or the Tri(Tn) problem, where Tri means that the boundary of these circles is a triangle, and Tn is the number of circles inside the triangle. In this paper, we propose an algorithm to solve the Tri(Tn) problem.<\/jats:p>","DOI":"10.3390\/a17080374","type":"journal-article","created":{"date-parts":[[2024,8,22]],"date-time":"2024-08-22T11:14:41Z","timestamp":1724325281000},"page":"374","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Extended General Malfatti\u2019s Problem"],"prefix":"10.3390","volume":"17","author":[{"given":"Ching-Shoei","family":"Chiang","sequence":"first","affiliation":[{"name":"Department of Computer and Information Science, Soochow University Taipei, Taiwan 100, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,22]]},"reference":[{"key":"ref_1","unstructured":"Fukagawa, H., and Pedoe, D. (1989). \u201cThe Malfatti Problem\u201d. Japanese Temple Geometry Problems (San GaKu), The Charles Babbage Research Centre."},{"key":"ref_2","first-page":"43","article-title":"The Malfatti Problem","volume":"1","author":"Bottema","year":"2000","journal-title":"Forum Geom."},{"key":"ref_3","first-page":"83","article-title":"Triangle centers associated with the Malfatti circles","volume":"3","year":"2003","journal-title":"Forum Geom."},{"key":"ref_4","unstructured":"Wolfram MathWorld (2024, August 15). Malfatti Circles. Available online: http:\/\/mathworld.wolfram.com\/MalfattiCircles.html."},{"key":"ref_5","unstructured":"Bernardin, L., Chin, P., DeMarco, P., Geddes, K.O., Hare, D.E.G., Heal, K.M., Labahn, G., May, J.P., McCarron, J., and Monagan, M.B. (1996). Maple Programming Guide, Maplesoft, a Division of Waterloo Maple Inc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"e103","DOI":"10.7717\/peerj-cs.103","article-title":"SymPy: Symbolic computing in Python","volume":"3","author":"Meurer","year":"2017","journal-title":"PeerJ Comput. Sci."},{"key":"ref_7","unstructured":"Kay, D.C. (1969). College Geometry, Holt, Rinehart and Winston."},{"key":"ref_8","unstructured":"(2024, August 15). Solution du Dernier des Deux Probl\u00e8mes Propos\u00e9s \u00e0 la Page 196 de ce Volume. Annales de Math\u00e9matiques Pures et Appliqu\u00e9es, Tome 1 (1810\u20131811); pp. 343\u2013348. Available online: http:\/\/www.numdam.org\/item\/AMPA_1810-1811__1__343_0\/."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1016\/j.comgeo.2010.06.005","article-title":"A generalized Malfatti problem","volume":"45","author":"Chiang","year":"2012","journal-title":"Comput. Geom."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/17\/8\/374\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:41:45Z","timestamp":1760110905000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/17\/8\/374"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,22]]},"references-count":9,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2024,8]]}},"alternative-id":["a17080374"],"URL":"https:\/\/doi.org\/10.3390\/a17080374","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2024,8,22]]}}}