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 orthocentroidal circle of a non-equilateral

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 ...

is the circle that has the triangle's

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 ...

and

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 ...

at opposite ends of its

diameter In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...

. This diameter also contains the triangle's

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 ...

and is a subset of the Euler line, which also 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 ...

outside the orthocentroidal circle. Andrew Guinand showed in 1984 that the triangle's

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 ...

must lie in the interior of the orthocentroidal circle, but not coinciding with the nine-point center; that is, it must fall in the open orthocentroidal disk punctured at the nine-point center... . The incenter could be any such point, depending on the specific triangle having that particular orthocentroidal disk. Furthermore, the Fermat point, the Gergonne point, and the symmedian point are in the open orthocentroidal disk punctured at its own center (and could be at any point therein), while the second Fermat point and Feuerbach point are in the exterior of the orthocentroidal circle. The set of potential locations of one or the other of the Brocard points is also the open orthocentroidal disk.. The square of the diameter of the orthocentroidal circle isAltshiller-Court, Nathan, ''College Geometry'', Dover Publications, 2007 (orig. Barnes & Noble 1952).

D^2-\tfrac(a^2+b^2+c^2),

where ''a, b,'' and ''c'' are the triangle's side lengths and ''D'' is the diameter of its

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 ...

References

External links

* {{cci Circles defined for a triangle