In geometry, the incircle of the medial triangle of a triangle is the Spieker circle, named after 19th-century German geometer Theodor Spieker. Its center, the Spieker center, in addition to being the incenter 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 balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...

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 hatchet. It is largely used as a kitchen or butcher knife and is mostly intended for splitting up large pieces of soft bones and slashing through ...

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, with around 183,000 inhabitants, largest city of the German state of Brandenburg. It is part of the Berlin/Brandenburg Metropolitan Region. Potsdam sits on the River Havel, a tributary of the Elbe, downstream of ...

, 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 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...

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

Construction

To find the Spieker circle of a triangle, the medial triangle 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''-dimen ...

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 to the circle within the medial triangle, creating the incircle. This circle center is named the Spieker center.

Nagel points and lines

Spieker circles also have relations to

Nagel point In geometry, the Nagel point (named for Christian Heinrich von Nagel) is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. It is the point of concurr ...

s. The incenter 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 surface of the figure. The same definition extends to any o ...

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 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 ...

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, Pa ...

, 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 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 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.

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