The Weighted Fermat Triangle Problem
We completely solve the generalized Fermat problem: given a triangle 𝑃1, 𝑃2, 𝑃3 and three positive numbers 𝜆1, 𝜆2, 𝜆3, find a point 𝑃 for which the sum 𝜆1𝑃1𝑃+𝜆2𝑃2𝑃+𝜆3𝑃3𝑃 is minimal. We show that the point always exists and is unique, and indicate necessary and sufficient conditions for the point to...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2008-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2008/283846 |
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| author | Yujin Shen Juan Tolosa |
| author_facet | Yujin Shen Juan Tolosa |
| author_sort | Yujin Shen |
| collection | DOAJ |
| description | We completely solve the generalized Fermat problem: given a triangle 𝑃1, 𝑃2, 𝑃3 and
three positive numbers 𝜆1, 𝜆2, 𝜆3, find a point 𝑃 for which the sum 𝜆1𝑃1𝑃+𝜆2𝑃2𝑃+𝜆3𝑃3𝑃
is minimal. We show that the point always exists and is unique, and indicate necessary
and sufficient conditions for the point to lie inside the triangle. We provide geometric
interpretations of the conditions and briefly indicate a connection with dynamical systems. |
| format | Article |
| id | doaj-art-3ca0baf7848347019ea785c5578653e4 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2008-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-3ca0baf7848347019ea785c5578653e42025-02-03T06:07:25ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252008-01-01200810.1155/2008/283846283846The Weighted Fermat Triangle ProblemYujin Shen0Juan Tolosa1Natural Sciences and Mathematics, The Richard Stockton College of New Jersey, Pomona, NJ 08240, USANatural Sciences and Mathematics, The Richard Stockton College of New Jersey, Pomona, NJ 08240, USAWe completely solve the generalized Fermat problem: given a triangle 𝑃1, 𝑃2, 𝑃3 and three positive numbers 𝜆1, 𝜆2, 𝜆3, find a point 𝑃 for which the sum 𝜆1𝑃1𝑃+𝜆2𝑃2𝑃+𝜆3𝑃3𝑃 is minimal. We show that the point always exists and is unique, and indicate necessary and sufficient conditions for the point to lie inside the triangle. We provide geometric interpretations of the conditions and briefly indicate a connection with dynamical systems.http://dx.doi.org/10.1155/2008/283846 |
| spellingShingle | Yujin Shen Juan Tolosa The Weighted Fermat Triangle Problem International Journal of Mathematics and Mathematical Sciences |
| title | The Weighted Fermat Triangle Problem |
| title_full | The Weighted Fermat Triangle Problem |
| title_fullStr | The Weighted Fermat Triangle Problem |
| title_full_unstemmed | The Weighted Fermat Triangle Problem |
| title_short | The Weighted Fermat Triangle Problem |
| title_sort | weighted fermat triangle problem |
| url | http://dx.doi.org/10.1155/2008/283846 |
| work_keys_str_mv | AT yujinshen theweightedfermattriangleproblem AT juantolosa theweightedfermattriangleproblem AT yujinshen weightedfermattriangleproblem AT juantolosa weightedfermattriangleproblem |