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Incircle


Incenter

An incircle is an inscribed circle of a polygon, i.e., a circle that is tangent to each of the polygon's sides. The center I of the incircle is called the incenter, and the radius r of the circle is called the inradius.

An incircle of a polygon is the two-dimensional case of an insphere of a solid.

While an incircle does not necessarily exist for arbitrary polygons, it exists and is moreover unique for triangles, regular polygons, and some other polygons including rhombi, bicentric polygons, and tangential quadrilaterals.

The incenter is the point of concurrence of the triangle's angle bisectors. In addition, the points M_A, M_B, and M_C of intersection of the incircle with the sides of DeltaABC are the polygon vertices of the pedal triangle taking the incenter as the pedal point (c.f. tangential triangle). This triangle is called the contact triangle.

The trilinear coordinates of the incenter of a triangle are 1:1:1.

The polar triangle of the incircle is the contact triangle.

IncircleNinePointCircle

The incircle is tangent to the nine-point circle.

Pedoe (1995, p. xiv) gives a geometric construction for the incircle.

TangentCirclesTriangle

There are four circles that are tangent to all three sides (or their extensions) of a given triangle: the incircle I and three excircles J_1, J_2, and J_3. These four circles are, in turn, all touched by the nine-point circle N.

The circle function of the incircle is given by

 l=-((-a+b+c)^2)/(4bc),
(1)

with an alternative trilinear equation given by

 alpha^2cos^4(1/2A)+beta^2cos^2(1/2B)+gamma^2cos^2(1/2C)-2betagammacos^2(1/2B)cos^2(1/2C)-2gammaalphacos^2(1/2C)-2alphabetacos^2(1/2A)cos^2(1/2B)=0
(2)

(Kimberling 1998, p. 40).

The incircle is the radical circle of the tangent circles centered at the reference triangle vertices.

Kimberling centers X_i lie on the incircle for i=11 (Feuerbach point), 1317, 1354, 1355, 1356, 1357, 1358, 1359, 1360, 1361, 1362, 1363, 1364, 1365, 1366, 1367, 2446, 2447, 3023, 3024, and 3025.

The area Delta of the triangle DeltaABC is given by

Delta=DeltaBIC+DeltaAIC+DeltaAIB
(3)
=1/2ar+1/2br+1/2cr
(4)
=1/2(a+b+c)r
(5)
=sr,
(6)

where s is the semiperimeter, so the inradius is

r=Delta/s
(7)
=sqrt(((s-a)(s-b)(s-c))/s).
(8)

Using the incircle of a triangle as the inversion center, the sides of the triangle and its circumcircle are carried into four equal circles (Honsberger 1976, p. 21).

IncircleConcurrence

Let a triangle DeltaABC have an incircle with incenter I and let the incircle be tangent to DeltaABC at T_A, T_C, (and T_B; not shown). Then the lines CI, T_AT_C, and the perpendicular to CI through A concur in a point P (Honsberger 1995).


See also

Circumcircle, Congruent Incircles, Congruent Incircles Point, Contact Triangle, Excircles, Incenter, Incentral Triangle, Inradius, Insphere, Japanese Theorem, Seven Circles Theorem, Tangent Circles, Tangential Triangle, Triangle Transformation Principle

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References

Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 53-55, 1888.Coxeter, H. S. M. and Greitzer, S. L. "The Incircle and Excircles." §1.4 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 10-13, 1967.Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., 1976.Honsberger, R. "An Unlikely Concurrence." §3.4 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 31-32, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 182-194, 1929.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Lachlan, R. "The Inscribed and the Escribed Circles." §126-128 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 72-74, 1893.Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., 1995.

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Incircle

Cite this as:

Weisstein, Eric W. "Incircle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Incircle.html

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