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Harmonic Conjugates (And lots of circles) (Because we really like circles) Math Club 12/05/2011. Free Powerpoint Templates. First Problem : Apollonius’s Problem. Your ship at point A is chasing a ship on point B.

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  1. Harmonic Conjugates (And lots of circles) (Because we really like circles) Math Club 12/05/2011 Free Powerpoint Templates

  2. First Problem: Apollonius’sProblem • Yourshipat point A ischasing a ship on point B. • Theirshipis travelling in some direction at constant speed in a line. • Yourshipis k times fasterthantheirship. • Yourship must alsosail in a straight line at constant speed. • Whichwayshouldyou go?

  3. Second Problem: Whatis a circle? • Καὶ παντὶ κέντρῳ καὶ διαστήματι κύκλον γράφεσθαι (a circle is defined by a center and a radius) – Euclid’s Third Postulate • Given A and B and a ratio k, a circleis the set of points P suchthat – Apollonius

  4. HarmonicConjugates • Is it possible to divide a line segment internally and externally in the same ratio? • Here, let’ssaythat. • Wesay C and D divideAB harmonically. • Then notice that. • So A and B divide CD harmonically as well. • AB and CD are harmonicconjugates.

  5. ConstructingHarmonicConjugates (1) • We have points A and B fixed, and have in mindsomenumber k. • Wewant a point P suchthat AP is k times BP. • Draw a circle of anylengtharound B. • Thendraw a circle k times as long around A. • Their intersection is a suitable P, if theyintersect.

  6. ConstructingHarmonicConjugates (2) • We have a point P now. • Draw the internalbisector of P and let C be the intersection with AB. • Draw the externalbisector of P and let D be the intersection. • AB and CD are harmonicconjugates!

  7. ConstructingHarmonicConjugates (3) • Now doesthisactuallywork? • By the angle bisectortheorem, . • By the external angle bisectortheorem, . • Hence. • So AB and CD are harmonicconjugates!

  8. Constructing the Circle of Apollonius • Notice that P is not part of this, sothisworks no matterwhat P wechoose. • Also notice that CPD is a right angle. • Then P has to lie on the circlewithdiameter CD!

  9. Apollonius, wesolvedyourproblem • Which direction do weneed to go? • Knowing AB, weconstruct the harmonicconjugates CD • Thenwedraw the circlewithdiameter CD • We know their direction, sothey are going to intersect the circleatsome point, say Q. • Wejustsailtowards Q!

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