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)

(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, 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, page 61:
  Lemma 4.9 (The Diameter of the Incircle). Let ${\\displaystyle ABC}$ be a triangle whose incircle is tangent to ${\\displaystyle {\\overline {BC}}}$ at ${\\displaystyle D}$ . If ${\\displaystyle {\\overline {DE}}}$ is a diameter of the incircle and ray ${\\displaystyle AE}$ meets ${\\displaystyle {\\overline {BC}}}$ at ${\\displaystyle X}$ , then ${\\displaystyle BD=CX}$ and ${\\displaystyle X}$ is the tangency point of the ${\\displaystyle A}$ -excircle to ${\\displaystyle {\\overline {BC}}}$ .
  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

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