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


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 be a triangle whose incircle is tangent to at . If is a diameter of the incircle and ray meets at , then and is the tangency point of the -excircle to .
      Incircles and excircles often have dual properties.

Usage notes[edit]

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

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