# excircle

## English

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

### 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, 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 ${\displaystyle ABC}$ 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).