excircle

plural excircles

1. geometryAn 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.

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

   1979, Dan Pedoe, Circles: A Mathematical View, published 1995, page 10

   2 more examples

   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.

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

   Lemma 4.9 (The Diameter of the Incircle). Let ABC be a triangle whose incircle is tangent to #92;overline#123;BC#125; at D. If #92;overline#123;DE#125; is a diameter of the incircle and ray AE meets #92;overline#123;BC#125; at X, then BD#61;CX and X is the tangency point of the A-excircle to #92;overline#123;BC#125;. Incircles and excircles often have dual properties.

   2016, Evan Chen, Euclidean Geometry in Mathematical Olympiads, page 61