Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...

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 ...

, the van Lamoen circle is a special

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 ...

associated with any given

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 ...

T

. It contains the

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 ...

s of the six triangles that are defined inside

T

by its three

median The median of a set of numbers is the value separating the higher half from the lower half of a Sample (statistics), data sample, a statistical population, population, or a probability distribution. For a data set, it may be thought of as the “ ...

s. Specifically, let

A

B

C

be the vertices of

T

, and let

G

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 figure. The same definition extends to any object in n-d ...

(the intersection of its three medians). Let

M_a

M_b

, and

M_c

be the midpoints of the sidelines

BC

CA

, and

AB

, respectively. It turns out that the circumcenters of the six triangles

AGM_c

BGM_c

BGM_a

CGM_a

CGM_b

, and

AGM_b

lie on a common circle, which is the van Lamoen circle of

T

History

The van Lamoen circle is named after the mathematician 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

X(1153)

in Clark Kimberling's comprehensive list of

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, ...

s. In 2003, Alexey Myakishev and Peter Y. Woo proved that the converse of the theorem is nearly true, in the following sense: let

P

be any point in the triangle's interior, and

AA'

BB'

, and

CC'

be its

cevian In geometry, a cevian is a line segment which joins a vertex of a triangle to a point on the opposite side of the triangle. Medians and angle bisectors are special cases of cevians. The name ''cevian'' comes from the Italian mathematician Giov ...

s, that is, the

line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...

s that connect each vertex to

P

and are extended until each meets the opposite side. Then the circumcenters of the six triangles

APB'

APC'

BPC'

BPA'

CPA'

, and

CPB'

lie on the same circle if and only if

P

is the centroid of

T

or its

orthocenter The orthocenter of a triangle, usually denoted by , is the point (geometry), point where the three (possibly extended) altitude (triangle), altitudes intersect. The orthocenter lies inside the triangle if and only if the triangle is acute trian ...

(the intersection of its three

altitudes Altitude 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 (e.g., aviation, geometry, geographical s ...

). A simpler proof of this result was given by Nguyen Minh Ha in 2005.

References

{{reflist, refs= {{mathworld, mode=cs2 , url = http://mathworld.wolfram.com/vanLamoenCircle.html , title = van Lamoen circle , access-date = 2014-10-10 {{citation , last = van Lamoen , first = Floor , year = 2000 , title = Problem 10830 , publisher = American Mathematical Monthly , volume = 107 , page = 893 (2002), ''Solution to Problem 10830''. American Mathematical Monthly, volume 109, pages 396-397. {{citation , last = Li , first = Kin Y. , year = 2001 , title = Concyclic problems , url = https://www.math.ust.hk/excalibur/v6_n1.pdf , journal = Mathematical Excalibur , volume = 6 , issue = 1 , pages = 1–2 {{citation , last1 = Myakishev , first1 = Alexey , last2 = Woo , first2 = Peter Y. , year = 2003 , title = On the Circumcenters of Cevasix Configuration , url = http://forumgeom.fau.edu/FG2003volume3/FG200305.pdf , journal = Forum Geometricorum , volume = 3 , pages = 57–63 {{citation , last = Ha , first = N. M. , year = 2005 , title = Another Proof of van Lamoen's Theorem and Its Converse , url = http://forumgeom.fau.edu/FG2005volume5/FG200516.pdf , journal = Forum Geometricorum , volume = 5 , pages = 127–132 {{citation , last = Kimberling , first = Clark , authorlink = Clark Kimberling , title = Encyclopedia of Triangle Centers , url = http://faculty.evansville.edu/ck6/encyclopedia/ETCPart2.html#X1153 , access-date = 2014-10-10. See ''X''(1153) = Center of the van Lemoen circle. Circles defined for a triangle

History

Properties

See also

References