---
OA_place: repository
OA_type: green
_id: '19494'
abstract:
- lang: eng
  text: Starting from any given rational-sided, right triangle, for example, the (3,4,5)-triangle
    with area 6, we use Euclidean geometry to show that there are infinitely many
    other rational-sided, right triangles of the same area. We show further that the
    set of all such triangles of a given area is finitely generated under our geometric
    construction. Such areas are known as “congruent numbers” and have a rich history
    in which all the results in this article have been proved and far more. Yet, as
    far as we can tell, this seems to be the first exploration using this kind of
    geometric technique.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Yik Tung
  full_name: Chan, Yik Tung
  id: c4c0afc8-9262-11ed-9231-d8b0bc743af1
  last_name: Chan
  orcid: 0000-0001-8467-4106
citation:
  ama: Chan S. Rational right triangles of a given area. <i>The American Mathematical
    Monthly</i>. 2018;125(8):689-703. doi:<a href="https://doi.org/10.1080/00029890.2018.1495491">10.1080/00029890.2018.1495491</a>
  apa: Chan, S. (2018). Rational right triangles of a given area. <i>The American
    Mathematical Monthly</i>. Taylor &#38; Francis. <a href="https://doi.org/10.1080/00029890.2018.1495491">https://doi.org/10.1080/00029890.2018.1495491</a>
  chicago: Chan, Stephanie. “Rational Right Triangles of a given Area.” <i>The American
    Mathematical Monthly</i>. Taylor &#38; Francis, 2018. <a href="https://doi.org/10.1080/00029890.2018.1495491">https://doi.org/10.1080/00029890.2018.1495491</a>.
  ieee: S. Chan, “Rational right triangles of a given area,” <i>The American Mathematical
    Monthly</i>, vol. 125, no. 8. Taylor &#38; Francis, pp. 689–703, 2018.
  ista: Chan S. 2018. Rational right triangles of a given area. The American Mathematical
    Monthly. 125(8), 689–703.
  mla: Chan, Stephanie. “Rational Right Triangles of a given Area.” <i>The American
    Mathematical Monthly</i>, vol. 125, no. 8, Taylor &#38; Francis, 2018, pp. 689–703,
    doi:<a href="https://doi.org/10.1080/00029890.2018.1495491">10.1080/00029890.2018.1495491</a>.
  short: S. Chan, The American Mathematical Monthly 125 (2018) 689–703.
date_created: 2025-04-05T10:51:16Z
date_published: 2018-09-28T00:00:00Z
date_updated: 2025-07-10T11:51:49Z
day: '28'
doi: 10.1080/00029890.2018.1495491
extern: '1'
external_id:
  arxiv:
  - '1706.05919'
intvolume: '       125'
issue: '8'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1706.05919
month: '09'
oa: 1
oa_version: Preprint
page: 689-703
publication: The American Mathematical Monthly
publication_identifier:
  eissn:
  - 1930-0972
  issn:
  - 0002-9890
publication_status: published
publisher: Taylor & Francis
quality_controlled: '1'
scopus_import: '1'
status: public
title: Rational right triangles of a given area
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 125
year: '2018'
...
---
_id: '909'
abstract:
- lang: eng
  text: We study the lengths of curves passing through a fixed number of points on
    the boundary of a convex shape in the plane. We show that, for any convex shape
    K, there exist four points on the boundary of K such that the length of any curve
    passing through these points is at least half of the perimeter of K. It is also
    shown that the same statement does not remain valid with the additional constraint
    that the points are extreme points of K. Moreover, the factor &amp;#xbd; cannot
    be achieved with any fixed number of extreme points. We conclude the paper with
    a few other inequalities related to the perimeter of a convex shape.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Arseniy
  full_name: Akopyan, Arseniy
  id: 430D2C90-F248-11E8-B48F-1D18A9856A87
  last_name: Akopyan
  orcid: 0000-0002-2548-617X
- first_name: Vladislav
  full_name: Vysotsky, Vladislav
  last_name: Vysotsky
citation:
  ama: Akopyan A, Vysotsky V. On the lengths of curves passing through boundary points
    of a planar convex shape. <i>The American Mathematical Monthly</i>. 2017;124(7):588-596.
    doi:<a href="https://doi.org/10.4169/amer.math.monthly.124.7.588">10.4169/amer.math.monthly.124.7.588</a>
  apa: Akopyan, A., &#38; Vysotsky, V. (2017). On the lengths of curves passing through
    boundary points of a planar convex shape. <i>The American Mathematical Monthly</i>.
    Mathematical Association of America. <a href="https://doi.org/10.4169/amer.math.monthly.124.7.588">https://doi.org/10.4169/amer.math.monthly.124.7.588</a>
  chicago: Akopyan, Arseniy, and Vladislav Vysotsky. “On the Lengths of Curves Passing
    through Boundary Points of a Planar Convex Shape.” <i>The American Mathematical
    Monthly</i>. Mathematical Association of America, 2017. <a href="https://doi.org/10.4169/amer.math.monthly.124.7.588">https://doi.org/10.4169/amer.math.monthly.124.7.588</a>.
  ieee: A. Akopyan and V. Vysotsky, “On the lengths of curves passing through boundary
    points of a planar convex shape,” <i>The American Mathematical Monthly</i>, vol.
    124, no. 7. Mathematical Association of America, pp. 588–596, 2017.
  ista: Akopyan A, Vysotsky V. 2017. On the lengths of curves passing through boundary
    points of a planar convex shape. The American Mathematical Monthly. 124(7), 588–596.
  mla: Akopyan, Arseniy, and Vladislav Vysotsky. “On the Lengths of Curves Passing
    through Boundary Points of a Planar Convex Shape.” <i>The American Mathematical
    Monthly</i>, vol. 124, no. 7, Mathematical Association of America, 2017, pp. 588–96,
    doi:<a href="https://doi.org/10.4169/amer.math.monthly.124.7.588">10.4169/amer.math.monthly.124.7.588</a>.
  short: A. Akopyan, V. Vysotsky, The American Mathematical Monthly 124 (2017) 588–596.
date_created: 2018-12-11T11:49:09Z
date_published: 2017-01-01T00:00:00Z
date_updated: 2025-07-10T12:01:35Z
day: '01'
department:
- _id: HeEd
doi: 10.4169/amer.math.monthly.124.7.588
ec_funded: 1
external_id:
  arxiv:
  - '1605.07997'
  isi:
  - '000413947300002'
intvolume: '       124'
isi: 1
issue: '7'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1605.07997
month: '01'
oa: 1
oa_version: Submitted Version
page: 588 - 596
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '291734'
  name: International IST Postdoc Fellowship Programme
publication: The American Mathematical Monthly
publication_identifier:
  issn:
  - 0002-9890
publication_status: published
publisher: Mathematical Association of America
publist_id: '6534'
quality_controlled: '1'
scopus_import: '1'
status: public
title: On the lengths of curves passing through boundary points of a planar convex
  shape
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 124
year: '2017'
...
---
_id: '4079'
acknowledgement: Research supported by Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862
article_processing_charge: No
article_type: original
author:
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Steven
  full_name: Skiena, Steven
  last_name: Skiena
citation:
  ama: Edelsbrunner H, Skiena S. On the number of furthest neighbor pairs in a point
    set. <i>American Mathematical Monthly</i>. 1989;96(7):614-618. doi:<a href="https://doi.org/10.1080/00029890.1989.11972250">10.1080/00029890.1989.11972250</a>
  apa: Edelsbrunner, H., &#38; Skiena, S. (1989). On the number of furthest neighbor
    pairs in a point set. <i>American Mathematical Monthly</i>. Mathematical Association
    of America. <a href="https://doi.org/10.1080/00029890.1989.11972250">https://doi.org/10.1080/00029890.1989.11972250</a>
  chicago: Edelsbrunner, Herbert, and Steven Skiena. “On the Number of Furthest Neighbor
    Pairs in a Point Set.” <i>American Mathematical Monthly</i>. Mathematical Association
    of America, 1989. <a href="https://doi.org/10.1080/00029890.1989.11972250">https://doi.org/10.1080/00029890.1989.11972250</a>.
  ieee: H. Edelsbrunner and S. Skiena, “On the number of furthest neighbor pairs in
    a point set,” <i>American Mathematical Monthly</i>, vol. 96, no. 7. Mathematical
    Association of America, pp. 614–618, 1989.
  ista: Edelsbrunner H, Skiena S. 1989. On the number of furthest neighbor pairs in
    a point set. American Mathematical Monthly. 96(7), 614–618.
  mla: Edelsbrunner, Herbert, and Steven Skiena. “On the Number of Furthest Neighbor
    Pairs in a Point Set.” <i>American Mathematical Monthly</i>, vol. 96, no. 7, Mathematical
    Association of America, 1989, pp. 614–18, doi:<a href="https://doi.org/10.1080/00029890.1989.11972250">10.1080/00029890.1989.11972250</a>.
  short: H. Edelsbrunner, S. Skiena, American Mathematical Monthly 96 (1989) 614–618.
date_created: 2018-12-11T12:06:49Z
date_published: 1989-01-01T00:00:00Z
date_updated: 2022-02-11T12:59:01Z
day: '01'
doi: 10.1080/00029890.1989.11972250
extern: '1'
intvolume: '        96'
issue: '7'
language:
- iso: eng
main_file_link:
- url: 'http://www.jstor.org/stable/2325182 '
month: '01'
oa_version: None
page: 614 - 618
publication: American Mathematical Monthly
publication_identifier:
  eissn:
  - 1930-0972
  issn:
  - 0002-9890
publication_status: published
publisher: Mathematical Association of America
publist_id: '2042'
quality_controlled: '1'
status: public
title: On the number of furthest neighbor pairs in a point set
type: journal_article
user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17
volume: 96
year: '1989'
...