excircle

English

The three excircles (orange) and incircle (blue) of a triangle (bold black)

The excircle of a quadrilateral

Etymology

ex- +‎ circle

Noun

excircle (plural excircles)

1. (geometry) An escribed circle; a circle outside a polygon (especially a triangle, but also sometimes a quadrilateral) that is tangent to each of the lines on which the sides of the polygon lie.
   • 1979, Dan Pedoe, Circles: A Mathematical View, published 1995, page 10:

     Also since the circle of inversion cuts both excircles orthogonally, each excircle inverts into itself.

   • 1999, Art Johnson, Famous Problems and Their Mathematicians, Teacher Ideas Press, page 174:

     Extend the sides of triangle QRS and construct the three excircles: One excircle is tangent to side QR and rays SQ and SR; one excircle is tangent to side SR and rays QS and QR; and one excircle is tangent to side SQ and rays RS and RQ.

   • 2016, Evan Chen, Euclidean Geometry in Mathematical Olympiads‎^[1], page 61:

     Lemma 4.9 (The Diameter of the Incircle). Let $ABC$ be a triangle whose incircle is tangent to ${\overline {BC}}$ at $D$. If ${\overline {DE}}$ is a diameter of the incircle and ray $AE$ meets ${\overline {BC}}$ at $X$, then $BD=CX$ and $X$ is the tangency point of the $A$-excircle to ${\overline {BC}}$.
     Incircles and excircles often have dual properties.

Usage notes

Any given triangle has exactly three excircles. A quadrilateral that has an excircle is said to be ex-tangential (or sometimes exscriptible).

Related terms

• excentre / excenter
• incircle

Translations

escribed circle

• Finnish: viereen piirretty ympyrä, vierusympyrä
• French: cercle exinscrit m
• Japanese: 傍接円 (ja) (bōsetsuen)
• Korean: 방접원 (ko) (bangjeobwon)
• Portuguese: círculo exinscrito m
• Swedish: utvändig tangeringscirkel c
• Vietnamese: đường tròn bàng tiếp

See also

• ex-tangential