In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the incenter of a triangle is a
triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal
angle bisectors of the triangle cross, as the point
equidistant
A point is said to be equidistant from a set of objects if the distances between that point and each object in the set are equal.
In two-dimensional Euclidean geometry, the locus of points equidistant from two given (different) points is th ...
from the triangle's sides, as the junction point of the
medial axis
The medial axis of an object is the set of all points having more than one closest point on the object's boundary. Originally referred to as the topological skeleton, it was introduced in 1967 by Harry Blum as a tool for biological shape recog ...
and innermost point of the
grassfire transform In image processing, the grassfire transform is the computation of the distance from a pixel to the border of a region. It can be described as "setting fire" to the borders of an image region to yield descriptors such as the region's skeleton or med ...
of the triangle, and as the center point of the
inscribed circle
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 incen ...
of the triangle.
Together with the
centroid
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
,
circumcenter, and
orthocenter, it is one of the four triangle centers known to the ancient Greeks, and the only one of the four that does not in general lie on the
Euler line. It is the first listed center, X(1), in
Clark Kimberling
Clark Kimberling (born November 7, 1942 in Hinsdale, Illinois) is a mathematician, musician, and composer. He has been a mathematics professor since 1970 at the University of Evansville. His research interests include triangle centers, integer ...
's
Encyclopedia of Triangle Centers
The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or "centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville.
, the ...
, and the
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
of the
multiplicative group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
*the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
of triangle centers.
[.][Encyclopedia of Triangle Centers](_blank)
, accessed 2014-10-28.
For
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...
s with more than three sides, the incenter only exists for
tangential polygons - those that have an incircle that is
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
to each side of the polygon. In this case the incenter is the center of this circle and is equally distant from all sides.
Definition and construction
It is a
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
in
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
that the three interior
angle bisectors of a triangle meet in a single point. In
Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
's
''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.
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
excircles
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.
...
of the given triangle. The incenter and excenters together form an
orthocentric system
In geometry, an orthocentric system is a set of four points on a plane, one of which is the orthocenter of the triangle formed by the other three. Equivalently, the lines passing through disjoint pairs among the points are perpendicular, and ...
.
The
medial axis
The medial axis of an object is the set of all points having more than one closest point on the object's boundary. Originally referred to as the topological skeleton, it was introduced in 1967 by Harry Blum as a tool for biological shape recog ...
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 image processing, the grassfire transform is the computation of the distance from a pixel to the border of a region. It can be described as "setting fire" to the borders of an image region to yield descriptors such as the region's skeleton or med ...
, in which one forms a continuous sequence of
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. 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.
Proofs
Ratio proof
Let the bisection of
and
meet at
, and the bisection of
and
meet at
, and
and
meet at
.
And let
and
meet at
.
Then we have to prove that
is the bisection of
.
In
,
, by the
Angle bisector theorem
In geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of ...
.
In
,
.
Therefore,
, so that
.
So
is the bisection of
Perpendicular proof
A line that is an angle bisector is equidistant from both of its lines when measuring by the perpendicular. At the point where two bisectors intersect, this point is perpendicularly equidistant from the final angle's forming lines (because they are the same distance from this angles opposite edge), and therefore lies on its angle bisector line.
Relation to triangle sides and vertices
Trilinear coordinates
The
trilinear coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...
for a point in the triangle give the ratio of distances to the triangle sides. Trilinear coordinates
for the incenter are given by
:
The collection of triangle centers may be given the structure of a
group under coordinatewise multiplication of trilinear coordinates; in this group, the incenter forms the
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
.
Barycentric coordinates
The
barycentric coordinates for a point in a triangle give weights such that the point is the weighted average of the triangle vertex positions.
Barycentric coordinates for the incenter are given by
:
where
,
, and
are the lengths of the sides of the triangle, or equivalently (using the
law of sines
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law,
\frac \,=\, \frac \,=\, \frac \,=\, 2R,
where , and ar ...
) by
:
where
,
, and
are the angles at the three vertices.
Cartesian coordinates
The
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
of the incenter are a weighted average of the coordinates of the three vertices using the side lengths of the triangle relative to the perimeter—i.e., using the barycentric coordinates given above, normalized to sum to unity—as weights. (The weights are positive so the incenter lies inside the triangle as stated above.) If the three vertices are located at
,
, and
, and the sides opposite these vertices have corresponding lengths
,
, and
, then the incenter is at
:
Distances to vertices
Denoting the incenter of triangle ''ABC'' as ''I'', the distances from the incenter to the vertices combined with the lengths of the triangle sides obey the equation
:
Additionally,
[. #84, p. 121.]
:
where ''R'' and ''r'' are the triangle's
circumradius
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every pol ...
and
inradius
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 incen ...
respectively.
Related constructions
Other centers
The distance from the incenter to the
centroid
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
is less than one third the length of the longest
median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic f ...
of the triangle.
By
Euler's theorem in geometry
In geometry, Euler's theorem states that the distance ''d'' between the circumcenter and incenter of a triangle is given by
d^2=R (R-2r)
or equivalently
\frac + \frac = \frac,
where R and r denote the circumradius and inradius respectively (the ...
, the squared distance from the incenter ''I'' to the circumcenter ''O'' is given by
[, p. 232.]
:
where ''R'' and ''r'' are the circumradius and the inradius respectively; thus the circumradius is at least twice the inradius, with equality only in the
equilateral case.
The distance from the incenter to the center ''N'' of the
nine point circle
In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are:
* The midpoint of ea ...
is
:
The squared distance from the incenter to the
orthocenter ''H'' is
:
Inequalities include:
:
The incenter is the
Nagel point of the
medial triangle (the triangle whose vertices are the midpoints of the sides) and therefore lies inside this triangle. Conversely the Nagel point of any triangle is the incenter of its
anticomplementary triangle
In Euclidean geometry, the medial triangle or midpoint triangle of a triangle is the triangle with vertices at the midpoints of the triangle's sides . It is the case of the midpoint polygon of a polygon with sides. The medial triangle is ...
.
The incenter must lie in the interior of a
disk whose diameter connects the centroid ''G'' and the
orthocenter ''H'' (the
orthocentroidal disk
In geometry, the orthocentroidal circle of a non-equilateral triangle is the circle that has the triangle's orthocenter and centroid at opposite ends of its diameter. This diameter also contains the triangle's nine-point center and is a sub ...
), but it cannot coincide with the
nine-point center, whose position is fixed 1/4 of the way along the diameter (closer to ''G''). Any other point within the orthocentroidal disk is the incenter of a unique triangle.
Euler line
The
Euler line of a triangle is a line passing through its
circumcenter,
centroid
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
, and
orthocenter, among other points.
The incenter generally does not lie on the Euler line; it is on the Euler line only for
isosceles triangle
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
s, for which the Euler line coincides with the symmetry axis of the triangle and contains all triangle centers.
Denoting the distance from the incenter to the Euler line as ''d'', the length of the longest median as ''v'', the length of the longest side as ''u'', the circumradius as ''R'', the length of the Euler line segment from the orthocenter to the circumcenter as ''e'', and the semiperimeter as ''s'', the following inequalities hold:
:
:
:
Area and perimeter splitters
Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter; every line through the incenter that splits the area in half also splits the perimeter in half. There are either one, two, or three of these lines for any given triangle.
Relative distances from an angle bisector
Let ''X'' be a variable point on the internal angle bisector of ''A''. Then ''X'' = ''I'' (the incenter) maximizes or minimizes the ratio
along that angle bisecter.
[Hajja, Mowaffaq, Extremal properties of the incentre and the excenters of a triangle", ''Mathematical Gazette'' 96, July 2012, 315-317.]
References
External links
*{{mathworld, id=Incenter, title=Incenter
Triangle centers