Apollonius Revisited Equidistance

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Apollonius Revisited: EquidistanceA Unified Approach To Tangency Problems Through the Employment of the ConicSectionsIntroduction &Motivation(A)Definition: DistanceDistance between:Two pointsA point and a lineA point and a circleTwo linesTwo circles(B)Definition: EquidistanceDefinitions:Perpendicular BisectorParabolaHyperbolaEllipse(C)Definition: EquilocusEquilocus of:Two pointsTwo linesA point and a lineA point and a circleA line and a circleTwo circles(D)Apollonius Tangency Problems: via Equidistance & EquilocusPPP, PPL, PPC, PLC, LLL, LLP, LLC, CCP, CCL, CCCApollonian GasketFurther ResearchAppendix:Constructing Parabola, Hyperbola, Ellipse in Geometer's SketchpadConstructing the intersection of two loci in Geometer's SketchpadConstructing the Apollonius Circle
Introduction and Motivation:Apollonius Revisited: EquidistanceA Unified Approach To Tangency ProblemsThrough the Employment of the Conic SectionsSelim Tezel 2011The approach to the Apollonius tangency problems described in this paper has beenmotivated by a desire to find a simple and unified approach to solving these challengingand beautiful problems. The extensive effort by 17 & 18th century geometers to solvethese problems posed by Apollonius (see Alfred Posamentier's Advanced EuclideanGeometry), employ a wide variety of construction ideas and techniques, which, whileingenious, in my opinion lack a certain aesthetics of simplicity as well as a unified insightinto the essential nature of these problems. It is important to note that in the attempt tofind a unified understanding of these problems I have abandoned the traditional approachto solve these problems with compass and straight-edge. Such work has already beendone by the geometers of the past and we know that they can be done. Instead, full powerof the dynamic visualization software Geometer's Sketchpad has been used to generateand explore interactions of loci relevant to the tasks at hand. My aim here is to explorethe tangency problems from the simple and unified perspective of the single concept of"equidistance". In the paper, definition of the word "Equilocus" has been given as the"middle path" between pairs of geometric objects. Below is a summary of some of theconclusions reached that are ultimately used in solving the Apollonian tangencyproblems:Object1 Object2 Equilocus: Middle PathPoint Point Perpendicular BisectorLine Line (intersecting) Angle BisectorPoint Line ParabolaCircle Line ParabolaCircle Point (Outside) HyperbolaCircle Circle (Outside) HyperbolaCircle Point (Inside) EllipseCircle Circle (Inside) EllipseGiven that so many books of Apollonius have been lost to history, though we will neverknow the truth, I have a strong feeling that the master himself was aware of theconnection of the conic sections (which he extensively explored in his Conics) to thetangency problems he posed.Acknowledgements:The work for this paper started upon watching a short and fun video called: "Doodling inMath Class: Infinity of Elephants" created by Vi Hart on the topic of the Gasket ofApollonius and I would like to express her my thanks for inspiring the initial thought forthis project. I also would like to thank all my former students in my Advanced Topics inMathematics: Theories of Infinity course, especially Woody Ahern, who helped mecorrect errors in the earlier drafts of this paper.
Page 1: Apollonius Revisited:EquidistanceA
Page 5 and 6: Distance between: A circle C and a
Page 7 and 8: (B) EQUIDISTANCEDefinition: A point
Page 9 and 10: Research: Construct and investigate
Page 11 and 12: Equilocus of:(a)Two intersecting li
Page 13 and 14: Equilocus of: A circle C and a poin
Page 15 and 16: Equilocus of: Two external circles
Page 17 and 18: Equilocus of: Two circles D inside
Page 19 and 20: (D) Apollonius Tangency Problems: E
Page 21 and 22: LLLTask: Construct a circle tangent
Page 23 and 24: LLCTask: Construct a circle tangent
Page 25 and 26: CCLTask: Construct a circle tangent
Page 27 and 28: Challenge: Study special cases not
Page 29 and 30: (c) a point and a hyperbolaAdvanced
Page 31 and 32: - Construct the perpendicular bisec
Page 33 and 34: -Hide all but the Focus, the direct
Page 35 and 36: -Construct a segment between M and
Page 37 and 38: - This tool will be handy if, say,
Page 39 and 40: Note: The difference between the di
Page 41 and 42: -Create a movable free point M on t
Page 43 and 44: -Hide all but the loci and two poin
Page 45 and 46: -Construct E and F, the intersectio

Apollonius Revisited Equidistance

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