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HandWiki. Incircle and Excircles of a Triangle. Encyclopedia. Available online: https://encyclopedia.pub/entry/34142 (accessed on 24 September 2026).
HandWiki. Incircle and Excircles of a Triangle. Encyclopedia. Available at: https://encyclopedia.pub/entry/34142. Accessed September 24, 2026.
HandWiki. "Incircle and Excircles of a Triangle" Encyclopedia, https://encyclopedia.pub/entry/34142 (accessed September 24, 2026).
HandWiki. (2022, November 11). Incircle and Excircles of a Triangle. In Encyclopedia. https://encyclopedia.pub/entry/34142
HandWiki. "Incircle and Excircles of a Triangle." Encyclopedia. Web. 11 November, 2022.
Incircle and Excircles of a Triangle
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The content is sourced from: https://handwiki.org/wiki/Incircle_and_excircles_of_a_triangle

In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides. The center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors. The center of an excircle is the intersection of the internal bisector of one angle (at vertex [math]\displaystyle{ A }[/math], for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex [math]\displaystyle{ A }[/math], or the excenter of [math]\displaystyle{ A }[/math]. Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the incircle together with the three excircle centers form an orthocentric system.:p. 182 All regular polygons have incircles tangent to all sides, but not all polygons do; those that do are tangential polygons. 

regular polygons polygons incircle

References

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  8. Johnson, Roger A., Advanced Euclidean Geometry, Dover, 2007 (orig. 1929).
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  11. Allaire, Patricia R.; Zhou, Junmin; and Yao, Haishen, "Proving a nineteenth century ellipse identity", Mathematical Gazette 96, March 2012, 161-165.
  12. Altshiller-Court, Nathan. College Geometry, Dover Publications, 1980.
  13. Posamentier, Alfred S., and Lehmann, Ingmar. The Secrets of Triangles, Prometheus Books, 2012.
  14. Franzsen, William N. (2011). "The distance from the incenter to the Euler line". Forum Geometricorum 11: 231–236. http://forumgeom.fau.edu/FG2011volume11/FG201126.pdf. .
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