Generalization of Apollonius Circle

Lecture Notes in Computer Science, 2001

Given a set of three circles in a plane, we want to find a circumcircle to these given circles called generators. This problem is well known as Apollonius Tenth Problem and is often encountered in geometric computations for CAD systems. This problem is also a core part of an algorithm to compute the Voronoi diagram of circles. We show that the problem can be reduced to a simple point-location problem among the regions bounded by two lines and two transformed circles. The transformed circles are produced from the generators via linear fractional transformations in a complex space. Then, some of the lines tangent to these transformed circles corresponds to the desired circumcircle to the generators. The presented algorithm is very simple yet fast. In addition, several degenerate cases are all incorporated into one single general framework.

Turkish J. Math, 2004

In this paper, we give a new characterization of Möbius transformations. To this end, a new concept of "Apollonius points of pentagons" is used.

2020

We investigate the analog of the circle of Apollonious in spherical geometry. This can be viewed as the pre-image through the stereographic projection of an algebraic curve of degree three. This curve consists of two connected components each being the “reflection” of the other through the center of the sphere. We give an equivalent equation for it, which is surprisingly, this time, of degree four.

awesomemath.org

This paper is on the Apollonian circles and isodynamic points of a triangle. Here we discuss some of the most intriguing properties of Apollonian circles and isodynamic points, along with several Olympiad problems, which can be solved using those properties.

2010

Given $\Delta ABC$ and angles $\alpha,\beta,\gamma\in(0,\pi)$ with $\alpha+\beta+\gamma=\pi$, we study the properties of the triangle $DEF$ which satisfies: (i) $D\in BC$, $E\in AC$, $F\in AB$, (ii) $\aangle D=\alpha$, $\aangle E=\beta$, $\aangle F=\gamma$, (iii) $\Delta DEF$ has the minimal area in the class of triangles satisfying (i) and (ii). In particular, we show that minimizer $\Delta DEF$, exists, is unique and is a pedal triangle, corresponding to a certain pedal point $P$. Permuting the roles played by the angles $\alpha,\beta,\gamma$ in (ii), yields a total of six such area-minimizing triangles, which are pedal relative to six pedal points, say, $P_1,....,P_6$. The main result of the paper is the fact that there exists a circle which contains all six points.

The main purpose of the paper is to present a new proof of the two celebrated theorems: one is " Ptolemy's Theorem " which explains the relation between the sides and diagonals of a cyclic quadrilateral and another is " Nine Point Circle Theorem " which states that in any arbitrary triangle the three midpoints of the sides, the three feet of altitudes, the three midpoints of line segments formed by joining the vertices and Orthocenter, total nine points are concyclic. Our new proof is based on a metric relation of circumcenter.

Turkish J. Math, 2004

In this paper, we give a new characterization of Möbius transformations. To this end, a new concept of "Apollonius points of pentagons" is used.

JOURNAL OF MECHANICS OF CONTINUA AND MATHEMATICAL SCIENCES

The concept of the circle has been known to human beings since before the beginning of recorded history. With the advent of the wheel, the study of the circle in detail played an important role in the field of science and technology. According to the author, there are three types of circles, 1) Countup circle, 2) Countdown circle, and 3) Point circle instead of two types of circles as defined by René Descartes in real plane coordinate geometry and Euler in the complex plane.

arXiv (Cornell University), 2023

We study some properties of a triad of circles associated with a triangle. Each circle is inside the triangle, tangent to two sides of the triangle, and externally tangent to the circle on the third side as diameter. In particular, we find a nice relation involving the radii of the inner and outer Apollonius circles of the three circles in the triad.

CIRCLE GEOMETRY SIMPLIFIED For Senior Secondary and Preliminary University students , 2021

My first motivation for writing this book came directly from a personal struggle faced during my secondary school days. The concept of circle geometry was completely a nightmare to me as I watched my teacher drew several circles and intersecting lines, within and without. Secondary, my teaching experience of over four years has taught me that several senior school students face similar challenges in the topic as I did during my days. I write this book to address these areas that pose a challenge to many a student who have eventually developed a phobia of the topic. Importantly also, this book is written to prepare final year students who will write external examinations such as the West African Examination Council (WAEC), National Examination Council (NECO), Unified Tertiary Matriculation Examination (UTME) and Post UTME as circle geometry happens to be one of the subjects that dominate these examinations. Preliminary and first year students at the University level can also find this book useful as it addresses some fundamental components of the curriculum. Potential engineering students can lay hold on this material at their preliminary stage, as it prepares them for important engineering components like technical drawing. To deal with this topic exhaustively, this book has been segmented into six chapters covering ten (10) fundamental theorems. Chapter one introduces circle geometry, chapter two treats theorems on chords, chapter three deals with segment theorems. Chapter four deals with cyclic quadrilateral theorems while chapter five handles tangent theorems. Chapter six treats a variety of questions cutting across all the ten theorems covered in this book. These questions and their solutions are purposely targeted at WAEC, NECO and UTME as majority of them are pulled from past questions of these examinations. The exercises at the end are meant to challenge students to familiarize themselves with the application of the theorems treated earlier. Answers are also provided for students to confirm their solutions.