Editing Incenter (section)

==Definition and construction==
It is a [[theorem]] in [[Euclidean geometry]] that the three interior [[angle bisector]]s of a triangle meet in a single point. In [[Euclid]]'s [[Euclid's Elements|''Elements'']], Proposition 4 of Book IV proves that this point is also the center of the inscribed circle of the triangle. The incircle itself may be constructed by dropping a perpendicular from the incenter to one of the sides of the triangle and drawing a circle with that segment as its radius.<ref>[[Euclid's Elements|Euclid's ''Elements]], [http://aleph0.clarku.edu/~djoyce/java/elements/bookIV/propIV4.html Book IV, Proposition 4: To inscribe a circle in a given triangle]. David Joyce, Clark University, retrieved 2014-10-28.</ref>

The incenter lies at equal distances from the three line segments forming the sides of the triangle, and also from the three lines containing those segments. It is the only point equally distant from the line segments, but there are three more points equally distant from the lines, the excenters, which form the centers of the [[Incircle and excircles of a triangle|excircles]] of the given triangle. The incenter and excenters together form an [[orthocentric system]].<ref>{{citation|last=Johnson|first=R. A.|title=Modern Geometry|publisher=Houghton Mifflin|location=Boston|year=1929|page=182}}.</ref>

The [[medial axis]] of a polygon is the set of points whose nearest neighbor on the polygon is not unique: these points are equidistant from two or more sides of the polygon. One method for computing medial axes is using the [[grassfire transform]], in which one forms a continuous sequence of [[Parallel curve|offset curves]], each at some fixed distance from the polygon; the medial axis is traced out by the vertices of these curves. In the case of a triangle, the medial axis consists of three segments of the angle bisectors, connecting the vertices of the triangle to the incenter, which is the unique point on the innermost offset curve.<ref>{{citation
 | last = Blum | first = Harry
 | editor-last = Wathen-Dunn | editor-first = Weiant
 | contribution = A transformation for extracting new descriptors of shape
 | location = Cambridge
 | quote = In the triangle three corners start propagating and disappear at the center of the largest inscribed circle
 | pages = 362–380
 | publisher = MIT Press
 | title = Models for the Perception of Speech and Visual Form
 | url = http://pageperso.lif.univ-mrs.fr/~edouard.thiel/rech/1967-blum.pdf
 | year = 1967}}.</ref> The [[straight skeleton]], defined in a similar way from a different type of offset curve, coincides with the medial axis for convex polygons and so also has its junction at the incenter.<ref>{{citation
 | last1 = Aichholzer | first1 = Oswin
 | last2 = Aurenhammer | first2 = Franz | author2-link = Franz Aurenhammer
 | last3 = Alberts | first3 = David
 | last4 = Gärtner | first4 = Bernd
 | doi = 10.1007/978-3-642-80350-5_65
 | issue = 12
 | journal = Journal of Universal Computer Science
 | mr = 1392429
 | pages = 752–761
 | title = A novel type of skeleton for polygons
 | url = http://www.jucs.org/jucs_1_12/a_novel_type_of
 | volume = 1
 | year = 1995}}.</ref>