The Spieker Circle and Certain Related Configurations

Author

• Berthola Elmo Lumzy, Xavier University of Louisiana

Date of Award

1952

Document Type

Thesis

Degree Name

Master of Arts (MA)

Department

Department of Mathematics

Keywords

Spieker Circle, Triangle Geometry, Nine-Point Circle, Synthetic Geometry, Euclidean Geometry, Median Triangle, Nagel Point, Homothetic Centers, Projective Geometry, Incircle

Abstract

This thesis, presented to the Department of Mathematics at Xavier University in partial fulfillment of requirements for the degree of Master of Arts in 1952, examines the Spieker circle, a circle inscribed in the median triangle of a given triangle, and extends its defining properties to related geometric configurations. Lumzy first establishes a series of analogies between the Spieker circle and the nine-point circle, demonstrating parallel relationships involving radius, center, harmonic conjugates, and homothetic centers relative to the incenter, median point, and Nagel point of a triangle.

The thesis then proves that the Spieker circle is inscribed in two congruent triangles and that its center coincides with the center of gravity of a triangle's perimeter. Building on a second definition of the Spieker circle stated in terms of harmonic conjugates, Lumzy introduces the ex-median points, ex-centers, and ex-Nagel points, points analogous to the median point, incenter, and Nagel point but located outside the triangle, and proves their collinearity as a key theorem. This result is used to define three ex-Spieker circles and to establish a series of theorems concerning the collinearity, congruency, and perspectivity of triangles formed from these extended configurations, culminating in results identifying the vertices of the original triangle as homothetic centers linking the Spieker and ex-Spieker circles. The thesis concludes by noting that its synthetic Euclidean approach left certain projective and analytical properties of these configurations, particularly harmonic properties of the outside configurations, open for further study.

Note: No original abstract was included; this abstract is provided by the XULA Digital Commons editors.

Recommended Citation

Lumzy, Berthola Elmo, "The Spieker Circle and Certain Related Configurations" (1952). Electronic Theses and Dissertations. 143.
https://digitalcommons.xula.edu/etd/143