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

incircle 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 the

medial triangle In Euclidean geometry, the medial triangle or midpoint triangle of a triangle is the triangle with vertices at the midpoints of the triangle's sides . It is the case of the midpoint polygon of a polygon with sides. The medial triangle is no ...

of a triangle is the Spieker circle, named after 19th-century German geometer Theodor Spieker. Its center, the

Spieker center In geometry, the Spieker center is a special point associated with a plane triangle. It is defined as the center of mass of the perimeter of the triangle. The Spieker center of a triangle is the center of gravity of a homogeneous wire frame in t ...

, in addition to being 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 ...

of the medial triangle, is the

center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the barycenter or balance point) is the unique point at any given time where the weight function, weighted relative position (vector), position of the d ...

of the uniform-density boundary of triangle. The Spieker center is also the point where all three

cleaver A cleaver is a large knife that varies in its shape but usually resembles a rectangular-bladed tomahawk. It is largely used as a kitchen knife, kitchen or butcher knife and is mostly intended for splitting up large pieces of soft bones and slas ...

s of the triangle (perimeter bisectors with an endpoint at a side's midpoint) intersect each other.

History

The Spieker circle and Spieker center are named after Theodor Spieker, a mathematician and professor from

Potsdam Potsdam () is the capital and largest city of the Germany, German States of Germany, state of Brandenburg. It is part of the Berlin/Brandenburg Metropolitan Region. Potsdam sits on the Havel, River Havel, a tributary of the Elbe, downstream of B ...

, Germany. In 1862, he published , dealing with planar geometry. Due to this publication, influential in the lives of many famous scientists and mathematicians including

Albert Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...

, Spieker became the mathematician for whom the Spieker circle and center were named.

Construction

To find the Spieker circle of a triangle, the

must first be constructed from the

midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dim ...

s of each side of the original triangle. The circle is then constructed in such a way that each side of the medial triangle is

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

to the circle within the medial triangle, creating the

. This circle center is named the

Nagel points and lines

Spieker circles also have relations to Nagel points. The

of the triangle and the Nagel point form a line within the Spieker circle. The middle of this line segment is the Spieker center. The Nagel line is formed by the incenter of the triangle, the Nagel point, and the

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 ...

of the triangle. The Spieker center will always lie on this line.

Nine-point circle and Euler line

Spieker circles were first found to be very similar to nine-point circles by Julian Coolidge. At this time, it was not yet identified as the Spieker circle, but is referred to as the "P circle" throughout the book. The nine-point circle with the Euler line and the Spieker circle with the Nagel line are analogous to each other, but are not

duals ''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers. Track listing :* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, ...

, only having dual-like similarities. One similarity between the nine-point circle and the Spieker circle deals with their construction. The nine-point circle is the circumscribed circle of the medial triangle, while the Spieker circle is the

inscribed An inscribed triangle of a circle In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figure G" means precisely the same th ...

circle of the medial triangle. With relation to their associated lines, the incenter for the Nagel line relates to the circumcenter for the Euler line. Another analogous point is the Nagel point and the othocenter, with the Nagel point associated with the Spieker circle and the orthocenter associated with the nine-point circle. Each circle meets the sides of the medial triangle where the lines from the orthocenter, or the Nagel point, to the vertices of the original triangle meet the sides of the medial triangle.

Spieker conic

The nine-point circle with the Euler line was generalized into the nine-point conic. Through a similar process, due to the analogous properties of the two circles, the Spieker circle was also able to be generalized into the Spieker conic. The Spieker conic is still found within the medial triangle and touches each side of the medial triangle, however it does not meet those sides of the triangle at the same points. If lines are constructed from each vertex of the medial triangle to the Nagel point, then the midpoint of each of those lines can be found. Also, the midpoints of each side of the medial triangle are found and connected to the midpoint of the opposite line through the Nagel point. Each of these lines share a common midpoint, S. With each of these lines reflected through S, the result is 6 points within the medial triangle. Draw a conic through any 5 of these reflected points and the conic will touch the final point. This was proven by de Villiers in 2006.

Spieker radical circle

The Spieker radical circle is the circle, centered at the Spieker center, which is orthogonal to the three excircles of the medial triangle.

References

* Dover reprint, 1960. *

External links

Spieker Conic and generalization of Nagel line
a

Generalizes Spieker circle and associated Nagel line. {{DEFAULTSORT:Spieker Circle Circles defined for a triangle