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HandWiki. Tangential Quadrilateral. Encyclopedia. Available online: https://encyclopedia.pub/entry/28565 (accessed on 10 October 2026).
HandWiki. Tangential Quadrilateral. Encyclopedia. Available at: https://encyclopedia.pub/entry/28565. Accessed October 10, 2026.
HandWiki. "Tangential Quadrilateral" Encyclopedia, https://encyclopedia.pub/entry/28565 (accessed October 10, 2026).
HandWiki. (2022, October 09). Tangential Quadrilateral. In Encyclopedia. https://encyclopedia.pub/entry/28565
HandWiki. "Tangential Quadrilateral." Encyclopedia. Web. 09 October, 2022.
Tangential Quadrilateral
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The content is sourced from: https://handwiki.org/wiki/Tangential_quadrilateral

In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the incircle of the quadrilateral or its inscribed circle, its center is the incenter and its radius is called the inradius. Since these quadrilaterals can be drawn surrounding or circumscribing their incircles, they have also been called circumscribable quadrilaterals, circumscribing quadrilaterals, and circumscriptible quadrilaterals. Tangential quadrilaterals are a special case of tangential polygons. Other less frequently used names for this class of quadrilaterals are inscriptable quadrilateral, inscriptible quadrilateral, inscribable quadrilateral, circumcyclic quadrilateral, and co-cyclic quadrilateral. Due to the risk of confusion with a quadrilateral that has a circumcircle, which is called a cyclic quadrilateral or inscribed quadrilateral, it is preferable not to use any of the last five names. All triangles can have an incircle, but not all quadrilaterals do. An example of a quadrilateral that cannot be tangential is a non-square rectangle. The section characterizations below states what necessary and sufficient conditions a quadrilateral must satisfy to be able to have an incircle.

quadrilateral quadrilaterals circumcircle

References

  1. Josefsson, Martin (2010), "Calculations concerning the tangent lengths and tangency chords of a tangential quadrilateral", Forum Geometricorum 10: 119–130, http://forumgeom.fau.edu/FG2010volume10/FG201013.pdf .
  2. Andreescu, Titu; Enescu, Bogdan (2006), Mathematical Olympiad Treasures, Birkhäuser, pp. 64–68 .
  3. Josefsson, Martin (2011), "More Characterizations of Tangential Quadrilaterals", Forum Geometricorum 11: 65–82, http://forumgeom.fau.edu/FG2011volume11/FG201108.pdf .
  4. Minculete, Nicusor (2009), "Characterizations of a Tangential Quadrilateral", Forum Geometricorum 9: 113–118, http://forumgeom.fau.edu/FG2009volume9/FG200910.pdf .
  5. Josefsson, Martin (2012), "Similar Metric Characterizations of Tangential and Extangential Quadrilaterals", Forum Geometricorum 12: 63–77, http://forumgeom.fau.edu/FG2012volume12/FG201207.pdf 
  6. Durell, C.V.; Robson, A. (2003), Advanced Trigonometry, Dover reprint, pp. 28–30 .
  7. Hajja, Mowaffaq (2008), "A condition for a circumscriptible quadrilateral to be cyclic", Forum Geometricorum 8: 103–106, http://forumgeom.fau.edu/FG2008volume8/FG200814.pdf .
  8. Siddons, A.W.; Hughes, R.T. (1929), Trigonometry, Cambridge Univ. Press, p. 203 .
  9. Grinberg, Darij, Circumscribed quadrilaterals revisited, 2008 http://www.cip.ifi.lmu.de/~grinberg/CircumRev.pdf
  10. Yiu, Paul, Euclidean Geometry, [1], 1998, pp. 156–157.
  11. Hoyt, John P. (1986), "Maximizing the Area of a Trapezium", American Mathematical Monthly 93 (1): 54–56, doi:10.2307/2322549 . https://dx.doi.org/10.2307%2F2322549
  12. Post at Art of Problem Solving, 2012 http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=466538
  13. Hoyt, John P. (1984), "Quickies, Q694", Mathematics Magazine 57 (4): 239, 242 .
  14. Josefsson, Martin (2010), "On the inradius of a tangential quadrilateral", Forum Geometricorum 10: 27–34, http://forumgeom.fau.edu/FG2010volume10/FG201005.pdf .
  15. Bogomolny, Alexander (2016), An Inradii Relation in Inscriptible Quadrilateral, Cut-the-knot, [2].
  16. Josefsson, Martin (2011), "When is a Tangential Quadrilateral a Kite?", Forum Geometricorum 11: 165–174, http://forumgeom.fau.edu/FG2011volume11/FG201117.pdf .
  17. Josefsson, Martin (2011), "The Area of a Bicentric Quadrilateral", Forum Geometricorum 11: 155–164, http://forumgeom.fau.edu/FG2011volume11/FG201116.pdf .
  18. Gutierrez, Antonio, "Circumscribed Quadrilateral, Diagonal, Chord, Proportion", [3], Accessed 2012-04-09.
  19. Josefsson, Martin (2010), "Characterizations of Bicentric Quadrilaterals", Forum Geometricorum 10: 165–173, http://forumgeom.fau.edu/FG2010volume10/FG201019.pdf .
  20. Myakishev, Alexei (2006), "On Two Remarkable Lines Related to a Quadrilateral", Forum Geometricorum 6: 289–295, http://forumgeom.fau.edu/FG2006volume6/FG200634.pdf .
  21. Dergiades, Nikolaos; Christodoulou, Dimitris M. (2017), "The two incenters of an arbitrary convex quadrilateral", Forum Geometricorum 17: 245–254, http://forumgeom.fau.edu/FG2017volume17/FG201727.pdf .
  22. "Ineq-G126 - Geometry - very nice!!!!", Post at Art of Problem Solving, 2011, [4]
  23. "Determine ratio OM/ON", Post at Art of Problem Solving, 2011 http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=455293
  24. Barton, Helen (1926), "On a circle attached to a collapsible four-bar", American Mathematical Monthly 33 (9): 462–465, doi:10.2307/2299611 . https://dx.doi.org/10.2307%2F2299611
  25. Bogomolny, Alexander, "When A Quadrilateral Is Inscriptible?", Interactive Mathematics Miscellany and Puzzles, [5].
  26. Chao, Wu Wei; Simeonov, Plamen (2000), "When quadrilaterals have inscribed circles (solution to problem 10698)", American Mathematical Monthly 107 (7): 657–658, doi:10.2307/2589133 . https://dx.doi.org/10.2307%2F2589133
  27. Vaynshtejn, I.; Vasilyev, N.; Senderov, V. (1995), "(Solution to problem) M1495", Kvant (6): 27–28. 
  28. Josefsson, Martin (2012), "Characterizations of Orthodiagonal Quadrilaterals", Forum Geometricorum 12: 13–25, http://forumgeom.fau.edu/FG2012volume12/FG201202.pdf .
  29. Hoehn, Larry (2011), "A new formula concerning the diagonals and sides of a quadrilateral", Forum Geometricorum 11: 211–212, http://forumgeom.fau.edu/FG2011volume11/FG201122.pdf .
  30. De Villiers, Michael (2011), "Equiangular cyclic and equilateral circumscribed polygons", Mathematical Gazette 95 (March): 102–107 .
  31. Bryant, Victor; Duncan, John (2010), "Wheels within wheels", The Mathematical Gazette 94 (November): 502–505 .
  32. Hess, Albrecht (2014), "On a circle containing the incenters of tangential quadrilaterals", Forum Geometricorum 14: 389–396, http://forumgeom.fau.edu/FG2014volume14/FG201437.pdf .
  33. Josefsson, Martin (2014), "The diagonal point triangle revisited", Forum Geometricorum 14: 381–385, http://forumgeom.fau.edu/FG2014volume14/FG201435.pdf .
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