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Center of the inscribed circle of a triangle
In geometry, 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
Incenter
Circles tangent to all three sides of a triangle
The center of the incircle is a triangle center called the triangle's incenter. An excircle or escribed circle of the triangle is a circle lying outside
Incircle_and_excircles
Theorem about inscribed and circumscribed circles
In geometry, the incenter–excenter lemma is the theorem that the line segment between the incenter and any excenter of a triangle, or between two excenters
Incenter–excenter_lemma
Polygon whose four sides all touch a circle
incircle of the quadrilateral or its inscribed circle, its center is the incenter and its radius is called the inradius. Since these quadrilaterals can be
Tangential_quadrilateral
Shape with three sides
angle in half. The three angle bisectors intersect in a single point, the incenter, which is the center of the triangle's incircle. The incircle is the largest
Triangle
Triangle center; intersection of all three of a triangle's splitters
The Nagel point, the centroid, and the incenter are collinear on a line called the Nagel line. The incenter is the Nagel point of the medial triangle;
Nagel_point
Division of something into two equal or congruent parts
triangle's incenter (the center of its incircle). There are either one, two, or three of these for any given triangle. A line through the incenter bisects
Bisection
4 planar points which are all orthocenters of triangles formed by the other 3
triangles formed from the four orthocentric points taken three at a time. The incenter of this common orthic triangle must be one of the original four orthocentric
Orthocentric_system
Convex, 4-sided shape with an incircle and a circumcircle
and centers of these circles are called inradius and circumradius, and incenter and circumcenter respectively. From the definition it follows that bicentric
Bicentric_quadrilateral
Geometric property of certain lines with respect to a given triangle
conjugate of the incenter of △ABC is the incenter itself. So the antiorthic axis, which is the central line associated with the incenter, is the axis of
Central_line_(geometry)
Center of mass of a triangle's perimeter
Spieker center of any triangle. The Spieker center of triangle △ABC is the incenter of the medial triangle of △ABC. That is, the Spieker center of △ABC is
Spieker_center
Property of points all lying on a single line
bases are collinear with the incenter. In a tangential trapezoid, the midpoints of the legs are collinear with the incenter. Pascal's theorem (also known
Collinearity
Polyhedron with four faces
tetrahedron is one in which the cevians that join the vertices to the incenters of the opposite faces are concurrent. An isogonic tetrahedron has concurrent
Tetrahedron
Triangle with at least two sides congruent
it follows that the Euler line coincides with the axis of symmetry. The incenter of the triangle also lies on the Euler line, something that is not true
Isosceles_triangle
Mathematical study of triangle properties (19th century–present)
Euclid's Elements contains description of the four special points – centroid, incenter, circumcenter and orthocenter - associated with a triangle. Even though
Modern_triangle_geometry
Point defined as a triangle center
and investigated by Schiffler et al. (1985). A triangle △ABC with the incenter I has its Schiffler point at the point of concurrence of the Euler lines
Schiffler_point
Geometric transformation applied to points with respect to a given triangle
conjugate P* for an arbitrary triangle △ABC. Angle bisectors (concur at incenter I) Lines from each vertex to P Lines to P reflected about angle bisectors
Isogonal_conjugate
Point in a triangle that can be seen as its middle under some criteria
in the middle of the triangle. For example, the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained
Triangle_center
Triangle with vertices at midpoints of another triangle's sides
the incenter of its reference triangle. In particular, this means that the incenter of a triangle must lie in its medial triangle. The incenter of the
Medial_triangle
Trapezoid whose sides are all tangent to the same circle
where I is the incenter. The angles ∠ AID and ∠ BIC in a tangential trapezoid ABCD, with bases AB and DC, are right angles. The incenter lies on the median
Tangential_trapezoid
Inscribed circle of a triangle's medial triangle
Theodor Spieker. Its center, the Spieker center, in addition to being the incenter of the medial triangle, is the center of mass of the uniform-density boundary
Spieker_circle
lines form the Euler-Gergonne-Soddy triangle. The Gergonne point and the incenter of the triangle are located on the Soddy line as well. The line is named
Soddy_line
Reflection of a triangle vertex's median over its angle bisector
Medians (concur at the centroid G) Angle bisectors (concur at the incenter I) Symmedians (concur at the symmedian point L)
Symmedian
does not go through its incenter unless the triangle is isosceles. For all non-isosceles triangles, the distance d from the incenter to the Euler line satisfies
List_of_triangle_inequalities
Online mathematics resource for cubic plane curves
AB, respectively The Neuberg cubic passes through the following points: incenter, circumcenter, orthocenter, both Fermat points, both isodynamic points
Catalogue_of_Triangle_Cubics
Triangle center: circumcenter of a triangle's excentral triangle
of a triangle. The Bevan point of a triangle is the reflection of the incenter across the circumcenter of the triangle. Bevan posed the problem of proving
Bevan_point
Line constructed from a triangle
Schiffler point, the Exeter point, and the Gossard perspector. However, the incenter generally does not lie on the Euler line; it is on the Euler line only
Euler_line
Lines which intersect at a single point
the triangle and bisecting the associated angle. They all meet at the incenter. Medians connect each vertex of a triangle to the midpoint of the opposite
Concurrent_lines
List of points considered center of a triangle
identified by an index number of the form X(n) —for example, X(1) is the incenter. The information recorded about each point includes its trilinear and barycentric
Encyclopedia of Triangle Centers
Encyclopedia_of_Triangle_Centers
On distance between centers of a triangle
Euler's theorem states that the distance d between the circumcenter and incenter of a triangle is given by d 2 = R ( R − 2 r ) {\displaystyle d^{2}=R(R-2r)}
Euler's_theorem_in_geometry
Triangle center associated with the nine-point circle
circle that cannot be the incenter is the nine-point center, and every other interior point of the circle is the incenter of a unique triangle. The distance
Nine-point_center
Intersection of the three symmedian lines of a triangle
(dotted) and symmedians (red). The symmedians intersect in the symmedian point L, the angle bisectors in the incenter I and the medians in the centroid G.
Lemoine_point
Point where the incircle and nine-point circle of a triangle are tangent
circle that is tangent to all three sides of the triangle. Its center, the incenter of the triangle, lies at the point where the three internal angle bisectors
Feuerbach_point
Orthocenter of a triangle's anticomplementary triangle
de Longchamp point is also collinear, along a different line, with the incenter and the Gergonne point of its triangle. The three circles centered at A
De_Longchamps_point
Circle tangent to two sides of a triangle and its circumcircle
incircle can be constructed with the following sequence of steps. Draw the incenter I {\displaystyle I} by intersecting angle bisectors. Draw a line through
Mixtilinear incircles of a triangle
Mixtilinear_incircles_of_a_triangle
Triangle center
detour property as well. The equal detour point, isoperimetric point, the incenter and the Gergonne point of a triangle are collinear, that is all four points
Equal_detour_point
Triangle center
triangle centers, unlike the classical triangle centers like centroid, incenter, and Steiner point. The Exeter point is defined as follows. Let △ABC be
Exeter_point
Size of a two-dimensional surface
triangle's area and its perimeter in half goes through the triangle's incenter (the center of its incircle). There are either one, two, or three of these
Area
Coordinate system based on distances from a triangle's sidelines
trilinear coordinates of the incenter of a triangle △ABC are 1 : 1 : 1; that is, the (directed) distances from the incenter to the sidelines BC, CA, AB
Trilinear_coordinates
Geometric objects with a common centre
Euler's theorem in geometry on the distance between the circumcenter and incenter of a triangle, two concentric circles (with that distance being zero) are
Concentric_objects
Intersection of triangle altitudes
The orthocenter is closer to the incenter I than it is to the centroid, and the orthocenter is farther than the incenter is from the centroid: H I ¯ < H
Orthocenter
Part of a line that is bounded by two distinct end points; line with two endpoints
those connecting various triangle centers to each other, most notably the incenter, the circumcenter, the nine-point center, the centroid and the orthocenter
Line_segment
Method of drawing geometric objects
internal angle bisectors, and its circumcenter, centroid, orthocenter, and incenter. These can be taken three at a time to yield 139 distinct nontrivial problems
Straightedge and compass construction
Straightedge_and_compass_construction
Geometric figure which circumscribes a circle
inradius. The vector from the incenter to the area centroid, GA , of a circumgonal region and the vector from the incenter to the centroid of its boundary
Circumgon
Triangle center
red arc in the center of the triangle is the locus of the Hofstadter r-points for 0 < r < 1. This locus passes through the incenter I of the triangle.
Hofstadter_points
Geometric formula for finding the ratio in which a line segment is divided by a point
a point internally or externally. It is used to find out the centroid, incenter and excenters of a triangle. In physics, it is used to find the center
Section_formula
Polyhedron formed by joining mirroring pyramids base-to-base
perpendicular line through the centroid of an arbitrary polygon or the incenter of a tangential polygon, depending on the source. Likewise, a right bipyramid
Bipyramid
Mean position of all the points in a shape
linear density, then the center of mass lies at the Spieker center (the incenter of the medial triangle), which does not (in general) coincide with the
Centroid
Trigonometric identity relating the sides and angles of a triangle
A triangle, showing the "incircle" and the partitioning of the sides. The angle bisectors meet at the incenter, which is the center of the incircle.
Law_of_cotangents
Unique curve associated with every triangle
rectangular hyperbola passing through important triangle centers such as the incenter, orthocenter, Gergonne point, Nagel point, mittenpunkt and Schiffler point
Feuerbach_hyperbola
Triangle containing a 90-degree angle
circle. The orthocenter lies on the circumcircle. The distance between the incenter and the orthocenter is equal to 2 r {\displaystyle {\sqrt {2}}r} . The
Right_triangle
Circle constructed from a triangle
orthocentroidal circle. Andrew Guinand showed in 1984 that the triangle's incenter must lie in the interior of the orthocentroidal circle, but not coinciding
Orthocentroidal_circle
Centers of the incircles of triangles inside a cyclic quadrilateral form a rectangle
□ABCD be an arbitrary cyclic quadrilateral and let M1, M2, M3, M4 be the incenters of the triangles △ABD, △ABC, △BCD, △ACD. Then the quadrilateral formed
Japanese theorem for cyclic quadrilaterals
Japanese_theorem_for_cyclic_quadrilaterals
Line segment from a midpoint of a triangle side which bisects its perimeter
center via cleavers. Reference triangle △ABC Angle bisectors (concur at incenter I) Cleavers (concur at Spieker center S) Medial triangle △DEF Inscribed
Cleaver_(geometry)
Equation of the form 1/a + 1/b = 1/c
quadrilateral, the inradius r, the circumradius R, and the distance x between the incenter and the circumcenter are related by Fuss' theorem according to 1 ( R −
Optic_equation
Triangle found by projecting a point onto the sides of another triangle
If P is the orthocenter, then △LMN is the orthic triangle. If P is the incenter, then △LMN is the intouch triangle. If P is the circumcenter, then △LMN
Pedal_triangle
Conic curves associated with a triangle
triangles ABC and A'B'C' Kiepert hyperbola showing the orthocenter, the incenter and the perpendicular asymptotes Kiepert parabola of triangle ABC. The
Kiepert_conics
Convex polygon that contains an inscribed circle
its internal angle bisectors are concurrent. This common point is the incenter (the center of the incircle). There exists a tangential polygon of n sequential
Tangential_polygon
Geometrical theorem relating the lengths of two segments that divide a triangle
been used to prove the following theorems/results: Coordinates of the incenter of a triangle Circles of Apollonius Alfred S. Posamentier: Advanced Euclidean
Angle_bisector_theorem
Cyclic polygon all of whose sides are tangent to an incircle
R, the inradius r, and the distance x between the circumcenter and the incenter. Some of these for specific n are: n = 5 : r ( R − x ) = ( R + x ) ( R
Bicentric_polygon
2012 book by Alfred S. Posamentier and Ingmar Lehmann
and fractals. Beyond the classical triangle centers (the circumcenter, incenter, orthocenter, and centroid) the book covers other centers including the
The_Secrets_of_Triangles
Number, approximately 2.41421
\left({\tfrac {1}{\sigma -1}}:{\tfrac {1}{\sigma +1}}\right)\sim (\sigma :1)} incenter, α = 3π/8 ( [ 1 + cos ( α ) ] − 1 : [ 1 + sec ( α ) ] − 1 ) ∼ ( sec
Silver_ratio
Circle derived from a triangle
identity. The centers of these circles are on the hyperbola through the incenter, three excenters, and X(49) (see below for X49). The homothetic centers
Sine-triple-angle_circle
triangle's area and its perimeter in half goes through the triangle's incenter. There can be one, two, or three of these for any given triangle. Area
Area_of_a_triangle
Shape with three equal sides
states that the distance t {\displaystyle t} between circumcenter and incenter is formulated as t 2 = R ( R − 2 r ) {\displaystyle t^{2}=R(R-2r)} . As
Equilateral_triangle
Concept in mathematics
{\displaystyle C} of any edge of the triangle (an "approximate" version of the incenter). A space is δ {\displaystyle \delta } -hyperbolic if every geodesic triangle
Hyperbolic_metric_space
Geometrical construction based on extending the sides of a triangle
Pa, Qa, Pb, Qb, Pc and Qc have all the same distance from the triangle incenter I, that is they lie on a common circle with center I. The radius of the
Conway_circle_theorem
Circle derived from a triangle
} which is also the distance between the circumcenter and incenter. Aside from the orthocenter the Fuhrmann circle intersects each altitude
Fuhrmann_circle
Line between midpoints of 2 diagonals in a 4-sided shape other than a parallelogram
triangle △ABP. If the quadrilateral is a tangential quadrilateral, then its incenter also lies on this line. Complete quadrangle Newton's theorem (quadrilateral)
Newton_line
2 points about which a triangle can be inverted into an equilateral triangle
unique isodynamic point, at its centroid(as well as its orthocenter, its incenter, and its circumcenter, which are concurrent); every non-equilateral triangle
Isodynamic_point
Coordinate system that is defined by points instead of vectors
^{2}-c^{2})&:&(a^{2}-b^{2}+c^{2})(-a^{2}+b^{2}+c^{2})\end{array}}} The incenter has coordinates a : b : c = sin α : sin β : sin γ . {\displaystyle
Barycentric_coordinate_system
Triangles without a right angle
which connects a vertex with the midpoint of the opposite side—and the incenter—the center of the circle that is internally tangent to all three sides—are
Acute_and_obtuse_triangles
Half of the sum of side lengths of a polygon
all the points on the triangle's edges. A line through the triangle's incenter bisects the perimeter if and only if it also bisects the area. A triangle's
Semiperimeter
Polynomial equation of degree 3
simply the triangle's incircle, its foci coincide with each other at the incenter, which lies on the real axis, and hence the derivative has duplicate real
Cubic_equation
Maps whose domain and codomain are acted on by the same group, and the map commutes
perimeter. However, triangle centers such as the centroid, circumcenter, incenter and orthocenter are not invariant, because moving a triangle will also
Equivariant_map
Polygon constructed from another
are concurrent at the circumcenter Angle bisectors are concurrent at the incenter The sums of the two pairs of opposite angles are equal The sums of the
Dual_polygon
Triangle center: symmedian point of the triangle's excentral triangle
connecting the centroid and the Gergonne point, the line connecting the incenter and the symmedian point and the line connecting the orthocenter with the
Mittenpunkt
Triangle area in terms of side lengths
prove Heron's formula, for example using trigonometry as below, or the incenter and one excircle of the triangle, or as a special case of De Gua's theorem
Heron's_formula
discovered that if you partition a triangle into 3 subtriangle with the incenter as a common point, then the Euler lines of those subtriangles and the original
Kurt_Schiffler
1972 US Congressional Medicare legislation
not differentiate payment based on dialysis method, location (home or incenter) or equipment used.[citation needed] The composite rate is intended to
End Stage Renal Disease Program
End_Stage_Renal_Disease_Program
In geometry, method for constructing a uniform polyhedron or plane tiling
polyhedron with Wythoff symbol a b|c. A vertex is placed so that it is on the incenter of ABC. This produces a polyhedron with Wythoff symbol a b c|. The vertex
Wythoff_construction
Study of angle-preserving transformations
triangle, that is, the nine-point center of the intouch triangle, the incenter and circumcenter of triangle ABC are collinear. Any two non-intersecting
Inversive_geometry
curve in this way, the middle point of the biarc should be chosen as the incenter of the triangle formed by the two endpoints of the Bézier curve and the
Biarc
Conic plane curve associated with a given triangle
Centers". Retrieved 11 October 2025. See X(11) = Feuerbach point, X(101) = Ψ(incenter, symmedian point), X(110) = Focus of Kiepert parabola, X(115) = Center
Triangle_conic
Geometric concept
through both Soddy centers, called the Soddy line, also passes through the incenter of the triangle, which is the homothetic center of the two Soddy circles
Soddy_circles_of_a_triangle
Quadrilateral whose vertices lie on a circle
the Poncelet point of its vertices. In a cyclic quadrilateral ABCD, the incenters M1, M2, M3, M4 (see the figure to the right) in triangles △DAB, △ABC,
Cyclic_quadrilateral
that is a line. This is the Simson line. If P {\displaystyle P} is the incenter of the triangle then the pedal circle is the incircle of the triangle and
Pedal_circle
Object modeling method
Compute the center of a circle or sphere enclosing an element of the mesh Incenter - Compute the center of a circle or sphere enclosed by an element of the
Polygonal_modeling
Medical intervention
week. Training can take from 2 to 8 weeks at which time one is dialyzed incenter, often in a separate home hemodialysis training unit. Introducing dialysis
Home_hemodialysis
Circle that passes through the vertices of a triangle
Euler's theorem in geometry, the distance between the circumcenter O and the incenter I is O I ¯ = R ( R − 2 r ) , {\displaystyle {\overline {OI}}={\sqrt {R(R-2r)}}
Circumcircle
Triangle center
triangle center X(175) of triangle △ABC. P lies on the line joining the incenter and the Gergonne point of △ABC. If P is an isoperimetric point of △ABC
Isoperimetric_point
Triangle center
the three excircles of the Yff central triangle of △ABC. Let I be the incenter of △ABC. Let D be the point on side BC such that ∠BID = ∠DIC, E a point
Yff_center_of_congruence
Triangle with integer side lengths
{\displaystyle abc{\big /}2(a+b+c).} Thus the squared distance between the incenter and the circumcenter of an integer triangle, given by Euler's theorem as
Integer_triangle
Formula for the area of a triangle
if and only if the vertex is on the opposite side of the line from the incenter. Then a a ′ + b b ′ + c c ′ = 2 K . {\displaystyle aa^{\prime }+bb^{\prime
Harcourt's_theorem
Theorem in Euclidean geometry
quadrilaterals, which shows that a rectangle is formed by the two pairs of incenters corresponding to the two possible triangulations of the quadrilateral
Japanese theorem for cyclic polygons
Japanese_theorem_for_cyclic_polygons
physics, typically the vector field I Luminous intensity, typically Iv the incenter of a triangle the electric current ionization energy, denoted I I represents:
Latin letters used in mathematics, science, and engineering
Latin_letters_used_in_mathematics,_science,_and_engineering
Notation for trigonometric relationships
}&=s^{2}-r^{2}-4rR\end{aligned}}} where R is the circumradius r is the incenter a b c = 2 S R {\displaystyle abc=2SR} s = a + b + c 2 {\displaystyle s={\frac
Conway_triangle_notation
Hyperbola constructed from a given triangle and point
describes a nine-point rectangular hyperbola passing through these centers: incenter X(1), the three excenters, the centroid X(2), the de Longchamps point X(20)
Nine-point_hyperbola
Obtuse triangle formed by the side and diagonals of a regular heptagon
OH=R{\sqrt {2}},} where R is the circumradius. The squared distance from the incenter I to the orthocenter is I H 2 = R 2 + 4 r 2 2 , {\displaystyle IH^{2}={\frac
Heptagonal_triangle
Special triangle based on arbitrary triangle
{\displaystyle R} its radius as well as I {\displaystyle I} denoting the incenter and r {\displaystyle r} its radius. Due to Euler's theorem one also has
Fuhrmann_triangle
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