In the

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

s, the

incircle 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 ...

and

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 each s ...

of a triangle are internally

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

to each other at the Feuerbach point of the triangle. The Feuerbach point is a

triangle center In geometry, a triangle center or triangle centre is a point in the triangle's plane that is in some sense in the middle of the triangle. For example, the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, ...

, meaning that its definition does not depend on the placement and scale of the triangle. It is listed as X(11) in Clark Kimberling'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. This resource is hosted at the University of Evansville The University of Evansville (UE) is a priv ...

, and is named after Karl Wilhelm Feuerbach..Encyclopedia of Triangle Centers
, accessed 2014-10-24. Feuerbach's theorem, published by Feuerbach in 1822, states more generally that the nine-point circle is tangent to the three

excircle 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. ...

s of the triangle as well as its incircle. A very short proof of this theorem based on Casey's theorem on the

bitangent In geometry, a bitangent to a curve is a line that touches in two distinct points and and that has the same direction as at these points. That is, is a tangent line at and at . Bitangents of algebraic curves In general, an algebraic cu ...

s of four circles tangent to a fifth circle was published by John Casey in 1866; Feuerbach's theorem has also been used as a test case for

automated theorem proving Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a majo ...

. The three points of tangency with the excircles form the Feuerbach triangle of the given triangle.

Construction

The

of a triangle ''ABC'' is a

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

that is tangent to all three sides of the triangle. Its center, the

incenter 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 or scale. The incenter may be equivalently defined as the point where the internal angle bis ...

of the triangle, lies at the point where the three internal angle bisectors of the triangle cross each other. The

is another circle defined from a triangle. It is so called because it passes through nine significant points of the triangle, among which the simplest to construct are the

midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dim ...

s of the triangle's sides. The nine-point circle passes through these three midpoints; thus, it is the

circumcircle In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertex (geometry), vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumrad ...

of the

medial 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 no ...

. These two circles meet in a single point, where they are

to each other. That point of tangency is the Feuerbach point of the triangle. Associated with the incircle of a triangle are three more circles, the excircles. These are circles that are each tangent to the three lines through the triangle's sides. Each excircle touches one of these lines from the opposite side of the triangle, and is on the same side as the triangle for the other two lines. Like the incircle, the excircles are all tangent to the nine-point circle. Their points of tangency with the nine-point circle form a triangle, the Feuerbach triangle.

Properties

The Feuerbach point lies on the line through the centers of the two tangent circles that define it. These centers are the

and

nine-point center In geometry, the nine-point center is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle. It is so called because it is the center of the nine-point circle, a circle ...

of the triangle. Let

x

y

, and

z

be the three distances of the Feuerbach point to the vertices of the

(the midpoints of the sides ''BC=a, CA=b'', and ''AB=c'' respectively of the original triangle). Then,Sa ́ndor Nagydobai Kiss, "A Distance Property of the Feuerbach Point and Its Extension", ''Forum Geometricorum'' 16, 2016, 283–290. http://forumgeom.fau.edu/FG2016volume16/FG201634.pdf :

x+y+z = 2\max(x,y,z),

or, equivalently, the largest of the three distances equals the sum of the other two. Specifically, we have

x=\frac, b-c, , \, y=\frac, c-a, , z=\frac, a-b, ,

where ''O'' is the reference triangle's

circumcenter In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcen ...

and ''I'' is its

. The latter property also holds for the tangency point of any of the excircles with the nine–point circle: the greatest distance from this tangency to one of the original triangle's side midpoints equals the sum of the distances to the other two side midpoints. If the incircle of triangle ABC touches the sides ''BC, CA, AB'' at ''X'', ''Y'', and ''Z'' respectively, and the midpoints of these sides are respectively ''P'', ''Q'', and ''R'', then with Feuerbach point ''F'' the triangles ''FPX'', ''FQY'', and ''FRZ'' are similar to the triangles ''AOI, BOI, COI'' respectively.

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 ...

for the Feuerbach point are :

1 - \cos (B - C) : 1 - \cos (C - A) : 1 - \cos (A - B).

Its barycentric coordinates are :

(s-a)(b-c)^2 : (s-b)(c-a)^2 : (s-c)(a-b)^2,

where ''s'' is the triangle's

semiperimeter In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name ...

s = \tfrac12(a+b+c)

. The three lines from the vertices of the original triangle through the corresponding vertices of the Feuerbach triangle meet at another triangle center, listed as X(12) in the Encyclopedia of Triangle Centers. Its trilinear coordinates are: :

1 + \cos (B - C) : 1 + \cos (C - A) : 1 + \cos (A - B).

Construction

Properties

Coordinates

References

Further reading