![Van Lamoen circle](https://upload.wikimedia.org/wikipedia/commons/5/59/Van_Lamoen_circle.svg)
In
Euclidean plane 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 ...
, the van Lamoen circle is a special
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
associated with any given
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 ...
. It contains the
circumcenter
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 polyg ...
s of the six triangles that are defined inside
by its three
medians.
[
Specifically, let , , be the vertices of , and let be its ]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 ...
(the intersection of its three medians). Let , , and be the midpoints of the sidelines , , and , respectively. It turns out that the circumcenters of the six triangles , , , , , and lie on a common circle, which is the van Lamoen circle of .[
]
History
The van Lamoen circle is named after the mathematician Floor van Lamoen https://nl.wikipedia.org/wiki/Floor_van_Lamoen who posed it as a problem in 2000.[ A proof was provided by Kin Y. Li in 2001,][ and the editors of the Amer. Math. Monthly in 2002.][
]
Properties
The center of the van Lamoen circle is point 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 comprehensive list of 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.[
In 2003, Alexey Myakishev and Peter Y. Woo proved that the converse of the theorem is nearly true, in the following sense: let be any point in the triangle's interior, and , , and be its ]cevian
In geometry, a cevian is a line that intersects both a triangle's vertex, and also the side that is opposite to that vertex. Medians and angle bisectors are special cases of cevians. The name "cevian" comes from the Italian mathematician Giovan ...
s, that is, the line segments that connect each vertex to and are extended until each meets the opposite side. Then the circumcenters of the six triangles , , , , , and lie on the same circle if and only if is the centroid of or its orthocenter
In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the '' ...
(the intersection of its three altitudes
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
).[ A simpler proof of this result was given by Nguyen Minh Ha in 2005.][
]
See also
* Parry circle
* Lester circle
References
[Eric W. Weisstein, ]
van Lamoen circle
' at Mathworld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Di ...
. Accessed on 2014-10-10.
[Floor van Lamoen (2000), ''Problem 10830'' American Mathematical Monthly, volume 107, page 893.]
[(2002), ''Solution to Problem 10830''. American Mathematical Monthly, volume 109, pages 396-397.]
[Kin Y. Li (2001), ]
Concyclic problems
'. Mathematical Excalibur, volume 6, issue 1, pages 1-2.
[Alexey Myakishev and Peter Y. Woo (2003), ]
On the Circumcenters of Cevasix Configuration
'. Forum Geometricorum, volume 3, pages 57-63.
[N. M. Ha (2005), ]
Another Proof of van Lamoen's Theorem and Its Converse
'. Forum Geometricorum, volume 5, pages 127-132.
[Clark Kimberling (), ]
', in the ''Encyclopedia of Triangle Centers'' Accessed on 2014-10-10.
Circles defined for a triangle