![Vectenpoints](https://upload.wikimedia.org/wikipedia/commons/a/a0/Vectenpoints.svg)
In the geometry of
triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
s, the Vecten points are two
triangle center
In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For exampl ...
s associated with any triangle. They may be constructed by constructing three squares on the sides of the triangle, connecting each square centre by a line to the opposite triangle point, and finding the point where these three lines meet. The outer and inner Vecten points differ according to whether the squares are extended outward from the triangle sides, or inward.
The Vecten points are named after an early 19th-century French mathematician named Vecten, who taught mathematics with
Gergonne
Joseph Diez Gergonne (19 June 1771 at Nancy, France – 4 May 1859 at Montpellier, France) was a French mathematician and logician.
Life
In 1791, Gergonne enlisted in the French army as a captain. That army was undergoing rapid expansion becau ...
in
Nîmes and published a study of the figure of three squares on the sides of a triangle in 1817.
[.]
Outer Vecten point
Let ABC be any given
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
. On the sides BC, CA, AB of the triangle, construct outwardly drawn three squares with centres
respectively. Then the lines
and
are concurrent. The point of concurrence is the outer Vecten point of the triangle ABC.
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 seq ...
'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 ...
, the outer Vecten point is denoted by X(485).
Inner Vecten point
Let ABC be any given
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
. On the sides BC, CA, AB of the triangle, construct inwardly drawn three squares respectively with centres
respectively. Then the lines
and
are concurrent. The point of concurrence is the inner Vecten point of the triangle ABC.
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 seq ...
'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 ...
, the inner Vecten point is denoted by X(486).
[
The line meets the ]Euler line
In geometry, the Euler line, named after Leonhard Euler (), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, includ ...
at the Nine point center of the triangle . The Vecten points lie on the Kiepert hyperbola *Friedrich Wilhelm August Ludwig Kiepert
Friedrich Wilhelm August Ludwig Kiepert (6 October 1846 – 5 September 1934) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typic ...
See also
* Napoleon points
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 ...
, a pair of triangle centers constructed in an analogous way using equilateral triangles instead of squares
References
External links
*{{mathworld, id=VectenPoints, title=Vecten Points
Triangle centers