Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

, the equal detour 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, ...

denoted by ''X''(176) 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 se ...

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 ...

. It is characterized by the equal detour property: if one travels from any vertex of a

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 ...

to another by taking a detour through some inner point , then the additional distance traveled is constant. This means the following equation has to hold:''Isoperimetric point and equal detour point''
at the

(retrieved 2020-02-07) :

=& \overline + \overline - \overline. \end

The equal detour point is the only point with the equal detour property

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

the following inequality holds for the angles of :M. Hajja, P. Yff: "The isoperimetric point and the point(s) of equal detour in a triangle". ''Journal of Geometry'', November 2007, Volume 87, Issue 1–2, pp 76–82, https://doi.org/10.1007/s00022-007-1906-y :

\tan\tfrac12\alpha + \tan\tfrac12\beta + \tan \tfrac12\gamma \leq 2

If the inequality does not hold, then the isoperimetric point possesses the equal detour property as well. The equal detour point, isoperimetric point, 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 ...

and the

Gergonne point 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 a triangle are

collinear In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...

, that is all four points lie on a common line. Furthermore, they form a

harmonic range In mathematics, a projective range is a set of points in projective geometry considered in a unified fashion. A projective range may be a projective line or a conic. A projective range is the dual of a pencil of lines on a given point. For instanc ...

(see graphic on the right).N. Dergiades
"The Soddy circles"
''

Forum Geometricorum ''Forum Geometricorum: A Journal on Classical Euclidean Geometry'' was a peer-reviewed open-access academic journal that specialized in mathematical research papers on Euclidean geometry. Founded in 2001, it was published by Florida Atlantic Unive ...

'' volume 7, pp. 191–197, 2007 The equal detour point is the center of the inner Soddy circle of a triangle and the additional distance travelled by the detour is equal to the diameter of the inner Soddy Circle. The barycentric coordinates of the equal detour point are :

\left( a+\frac : b+\frac : c+\frac \right).

and 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 ...

are:

1 + \frac \ :\ 
1 + \frac \ :\ 
1 + \frac

References

External links

{{commons category, Equal detour point
isoperimetric and equal detour points
– interaktive Illustration auf Geogebratube Triangle centers