"Incircle" redirects here. For incircles of non-triangle polygons, see Tangential quadrilateral and Tangential polygon.
Incircle and excircles of a triangle.
Extended sides of triangle △ABC
Incircle (incenter at I)
Excircles (excenters at JA, JB, JC)
Internal angle bisectors
External angle bisectors (forming the excentral triangle)
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.[1]
An excircle or escribed circle[2] of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides.[3]
The center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors.[3][4] The center of an excircle is the intersection of the internal bisector of one angle (at vertex A, for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex A, or the excenter of A.[3] Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the incircle together with the three excircle centers form an orthocentric system.[5] but not all polygons do; those that do are tangential polygons. See also tangent lines to circles.
^Kay (1969, p. 140)
^Altshiller-Court (1925, p. 74)
^ abcAltshiller-Court (1925, p. 73)
^Kay (1969, p. 117)
^Johnson 1929, p. 182.
and 25 Related for: Incircle and excircles information
to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. An excircle or escribed circle of the triangle is...
that the nine-point circle is tangent to the three excircles of the triangle as well as its incircle. A very short proof of this theorem based on Casey's...
-mixtilinear incircle. Every triangle has three unique mixtilinear incircles, one corresponding to each vertex. The A {\displaystyle A} -excircle of triangle...
Great circle Great-circle distance Circle of a sphere Horocycle Incircleandexcircles of a triangle Inscribed circle Johnson circles Magic circle (mathematics)...
on the tangency of the nine-point circle of a triangle with its incircleandexcircles Descartes' theorem Ford circle Bankoff circle Archimedes' twin circles...
Circle List of circle topics Thales' theorem Circumcircle Concyclic Incircleandexcircles of a triangle Orthocentric system Monge's theorem Power center Nine-point...
Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly from the 5th century...
the radii of the incircleand the three excircles: a + b + c = r + r a + r b + r c . {\displaystyle a+b+c=r+r_{a}+r_{b}+r_{c}.} Acute and obtuse triangles...
Εὐκλείδης; fl. 300 BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the...
to the incircleandexcircles of a triangle. Terquem's other contributions to mathematics include naming the pedal curve of another curve, and counting...
of a given circle by using only a finite number of steps with a compass and straightedge. The difficulty of the problem raised the question of whether...
orthocentric points. The common nine-point circle is tangent to all 16 incirclesandexcircles of the four triangles whose vertices form the orthocentric system...
solutions to the general LLL problem (the incircleandexcircles of the triangle formed by the three lines). Points and lines may be viewed as special cases...
titled Volume I, From Thales to Euclid and Volume II, From Aristarchus to Diophantus. It got positive reviews and is still used today. Ten years later,...
with its incircle. The extouch triangle of a reference triangle has its vertices at the points of tangency of the reference triangle's excircles with its...
for all Heronian triangles, but additionally the centers of the incircleandexcircles are at lattice points.: Thm. 5 See also Integer triangle § Heronian...
triangle, and its center is the center of mass of the perimeter of the triangle.) Any vertex, the tangency of the opposite side with the incircle, and the Gergonne...
result is that the incircles can be exchanged for the excircles to the same triangles (tangent to the sides of the quadrilateral and the extensions of...
the plane of the reference triangle and associated with it in some way. For example, the circumcircle and the incircle of the reference triangle are triangle...
semiperimeter also applies to tangential quadrilaterals, which have an incircleand in which (according to Pitot's theorem) pairs of opposite sides have...
incircle. Further important inellipses are the Steiner inellipse, which touches the triangle at the midpoints of its sides, the Mandart inellipse and...
In geometry, the incircle of the medial triangle of a triangle is the Spieker circle, named after 19th-century German geometer Theodor Spieker. Its center...
those that have an incircle that is tangent to each side of the polygon. In this case the incenter is the center of this circle and is equally distant...
circle of infinite radius. In every triangle a unique circle, called the incircle, can be inscribed such that it is tangent to each of the three sides of...
Global Information