Schlick Circle

Schlick Circle
Name	Schlick Circle
Type	Circle
Related	Euler line, Nine-point circle, Incircle, Circumcircle, Centroid (geometry)

Schlick Circle The Schlick Circle is a specific Euclidean circle associated to a reference triangle studied in triangle geometry. It arises in constructions related to the Euler line, Nagel point, Gergonne point, and other classical centers such as the Orthocenter (triangle), Circumcenter (triangle), and Incenter (triangle). The circle has well-defined metric relations to the Nine-point circle, Spieker circle, and the Appolonius problem loci for given ratio constraints.

Definition and basic properties

The Schlick Circle is defined for a nondegenerate triangle with vertices often denoted by the triangle's standard labels used in classical sources such as works by Euler, Gauss, Lemoine, Brocard, and Weber. It passes through or is tangent to notable points including the Nagel point, Gergonne point, and intersections of cevians studied by Ceva's theorem and Menelaus' theorem. Its radius, center, and chordal relationships are invariant under similarity transformations and transform predictably under affine maps that send triangles to triangles respecting barycentric coordinates used by Routh and trilinear coordinate representations.

Construction and geometric characterization

Classical constructions of the Schlick Circle use compass-and-straightedge steps reminiscent of procedures by Euclid, Apollonius, and later expositors such as Poncelet and Steiner. One construction fixes three special points derived from intersections of the internal bisector and external bisector lines that meet at the Incenter (triangle) and excenters, then constructs the circumcircle through those points analogous to the procedure for the Mandart circle or the Feuerbach circle. Another characterization presents the Schlick Circle as the locus of points P for which directed power differences to pairs of reference points match linear forms studied by Carnot and Steiner in relation to pedal circles and pedal triangles.

Relationship to other notable circles and points

The Schlick Circle is intimately related to the Nine-point circle, sharing homothety centers with the line containing the Euler line and certain homothetic images that involve the Centroid (geometry), Orthocenter (triangle), and Circumcenter (triangle). It may be tangent to the Incircle or an excircle in special triangle families studied by Morley and Brocard analysts. Its intersections with the Circumcircle and the Jerabek hyperbola yield points connected to the Isogonal conjugate operation and the symmedian point, linking it to work by Clark Kimberling and catalogs of triangle centers.

Algebraic equations and coordinate representations

In barycentric coordinates relative to triangle vertices A, B, C, the Schlick Circle admits an equation expressible as a homogeneous quadratic derived from squared side-lengths and the area expression Heron attributed to Heron. Trilinear coordinate forms involve distances to sidelines and linear combinations akin to formulas used by Möbius and Ceva-type algebraic manipulations. In Cartesian coordinates with the reference triangle placed at convenient positions related to Descartes and affine normalization, the circle equation reduces to x^2 + y^2 + ux + vy + w = 0 with coefficients determined by Law of Cosines relations credited to Euclid and Ptolemy-style cyclic quadrilateral algebra. Determinantal representations use matrices analogous to those in Sylvester's determinant theorem and Cayley–Menger determinant formulations for circle and sphere conditions.

Historical background and naming

The circle is named in honor of an investigator whose surname appears in modern triangle-center literature and who contributed to triangle center catalogs alongside figures like Euler, Routh, Lemoine, and Brocard. The development of its properties drew on classical threads from Euclid, Apollonius, and Poncelet and was later systematized in the 19th and 20th centuries through the efforts of geometers such as Carnot, Steiner, and modern compilers like Clark Kimberling and cataloguers of triangle centers at institutions including university geometry groups in Paris, Berlin, and Princeton.

Applications and examples in Euclidean geometry

The Schlick Circle appears in problem sets and olympiad-style configurations that reference classical constructions by Euclid, Archimedes, and modern competitors in International Mathematical Olympiad problems. It serves as a test case for transformation techniques involving inversion, homothety, and Isogonal conjugation used by authors such as R. A. Johnson and in treatises from Oxford and Cambridge on advanced Euclidean geometry. Explicit examples include scalene, isosceles, and equilateral triangles where the Schlick Circle reduces to well-known circles like the Circumcircle or the Nine-point circle studied by Euler and Feuerbach, and instructive loci yielding pedagogical demonstrations for courses at Harvard and MIT.

Category:Circle (geometry)