geometry of complex numbers

RISTO MALCHESKI SAVA GROZDEV KATERINA ANEVSKA

GEOMETRY OF COMPLEX NUMBERS

2015 

Reviewers: Prof. Dr. Lidia Ilievska, Assoc. Prof. Dr. Veselin Nenkov

© Arhimed, Publisher, 2015 © Risto Malcheski, Sava Grozdev, Katerina Anevska, Authors, 2015 © Angelina Avramova, Graphic design, 2015 ISBN 978-954-799-



Contents Preface ....................................................................................................... 5

CHAPTER I COMPLEX NUMBERS 1. 2. 3. 4. 5. 6. 7. 8. 9.

The concept of complex number, basic properties ....................... 7 Algebraic notation of a complex number .................................. 10 A conjugate complex number ..................................................... 11 Geometric presentation of a complex number . .......................... 16 Extended complex plane. Reimann interpretation of complex number ..................................................................... 19 Trigonometric entry of a complex number ................................ 22 Roots of a complex number ........................................................ 24 Exponential entry of a complex number . ................................... 28 The set Cn. .................................................................................. 32

CHAPTER II TRANSFORMATION IN EUCLEDIAN PLANE 1. Line equation. Parallel and perpendicular lines . ........................ 36 2. Distance from a point to a line . .................................................. 40 3. Circle equation . ......................................................................... 42 4. Direct similarities . ...................................................................... 46 5. Motions ....................................................................................... 51 6. Homothety .................................................................................. 56 7. Indirect similarity . ...................................................................... 62 8. Inversion .................................................................................... 70 9. Mőbius transformation . .............................................................. 80 10. Geometric properties of a Mőbius transformation . .................... 83

CHAPTER III GEOMETRY OF CIRCLE AND TRIANGLE 1. 2. 3. 4. 5. 6.

Central and inscribed angles ....................................................... 89 Power of a point with respect to a circle...................................... 91 Radical axis and radical center ................................................... 95 A pencil and a bundle of circles . ................................................ 99 Orthocenter and centroid of a triangle ..................................... 103 Right angled triangle . ............................................................... 109 

7. Euler line and Euler circle ........................................................ 111 8. Menelaus’ theorem . ................................................................. 114 9. Pascal’s and Desargues’ theorem .............................................. 116 10. Triangular coordinates .............................................................. 118 11. Ceva’s and Van Aubel’s theorem .............................................. 120 12. Area of a triangle ...................................................................... 123 13. Incircles and excircles of a triangle ......................................... 127 14. Stewart’s theorem ..................................................................... 133 15. Simpson line.............................................................................. 136 16. Ptolemy’s theorem .................................................................... 138 17. Inner product . ........................................................................... 140

CHAPTER IV EXAMPLES AND EXERCISES 1. 2. 3. 4.

Examples (Chapter I) ................................................................ 144 Exercises (Chapter I) ................................................................ 159 Examples (Chapter II, Chapter III) ........................................... 164 Exercises (Chapter II, Chapter III) ........................................... 222

References . ............................................................................................ 236



PREFACE

No research of people could be named true science if it is not supported by a mathematical proof. Reliability of the assertions in different subjects is problematic when application of any mathematical domain is missing, i. e. when there is no connection with mathematics. Leonardo da Vinci

The present book is devoted to students of the last school grades, university students, teachers, lecturers and all lovers of mathematics who want to enrich their knowledge and skills in complex numbers and their numerous applications in Euclidean Geometry. Few countries in the world include complex numbers in their secondary school curriculum but even if included the volume of the corresponding content is quite insufficient consisting of elementary operations and geometric representation at most. The significance of the complex numbers is far from a real recognition in known textbooks and scholar literature. The applications not only in mathematics but also in many other subjects are considerable and the present book is a strong proof of such a statement. Mainly, the book will be useful for outstanding students with high potentialities in mathematics preparing themselves for successful participation in mathematical competitions and Olympiads. Other target groups are not excluded, namely those, whose representatives like to meet real challenges, connected with unexpected circumstances in problem solving. The material in the book is divided into four chapters. The first one contains basic properties of the complex numbers, their algebraic notation, the notion of a conjugate complex number, geometric, trigonometric and exponential presentations, also interesting facts in connection with Reimann interpretation and the set Cn. The second chapter includes various transformations of complex numbers in the Euclidean plane like similarity, homothety, inversion and Mőbius transformation. The third chapter is dedicated to the geometry of circle and triangle on the base of complex numbers. Numerous theorems are proposed, namely: Menelau’s theorem, Pascal’s and Desargue’s theorem, Ceva’s and Van Aubel’s theorem, Stewart’s theorem, Ptolemy’s theorem and others. Exercises and problems are included in the Fourth chapter: 122 examples with solutions and 161 solved problems pare proposed. Together with all the 138 theorems, lemmas and corollaries accompanied by 64 examples and 88 figures, the book turns out to be a rather exhaustive collection of the complex number applications in Euclidean Geometry.



A high just appraisal of the book is due to the numerous non-standard problems in it taken from the National Olympiads of Bulgaria, China, Iran, Japan, Korea, Poland, Romania, Russia, Serbia, Turkey, Ukraine and others but also from Several International and Balkan Mathematical Olympiads. The authors express their sincere thanks to the Editorial House “Archimedes 2” for the decision to accept the manuscript and to support the appearance of the present book. Also, sincere thanks to the reviewers Prof. Dr. Lidia Ilievska and Assoc. Prof. Dr. Veselin Nenkov for their helpful criticism, removal of mistakes and well-wishing advices, which contributed to the final quality of the book. Of course, different lapses are possible and we will be grateful to the readers in case they notice such and bring them to the attention of the Editor.

February, 2015 Skopje and Sofia



The authors

REFERENCES 1. Alfors, L. V.: Complex analysis, second edition, New-York-St. Louis-San Fran­cis­coToronto-London-Sydney, 1966 2. Anevska, K., Gogovska, V., Malcheski, R.: The role of complex numbers in interdisciplinary integration in mathematics teaching, Elsevier, Procedia - Social and Behavioral Sciences, 2014 3. Anevska, K.: Methodical approach for introduction of an exponential entry of complex numbers in secondary education, Proc. V Congres of SMM, Ohrid, September, 2014 4. Arslanagić, Š.: Matematika za nadarene (drugo izdanje), Bosanska riječ, Sarajevo, 2005 5. Ašić, M. i dr.: Međunarodne matematičke olimpijade, DM Srbije, Beograd, 1986 6. Conway, J. B.: Functions of One Complex Variable, Springer-Werlag, New-YorkHaidelberg-Berlin, 1987 7. Đurković, R.: Matematička takmičenja srednjoškolaca u Jugoslaviji 1990, DM Srbije, Beograd, 1991 8. Engel, A.: Problem-Solving Strategies, Springer-Verlag, New York-BerlinHeidelberg, 1997 9. Govedarica, V.: Matematička takmičenja u Republici Srpskoj, Zavod za udžbenike I nastavna sredstva, Istočno Sarajevo, 2007 10. Grozdev, S., Nenkov, V.: Design of Information-educational Environment for the Establishment of New Facts in Mathematics, Procedia – Social and Behavioral Sciences, volume 128, (22 April 2014), International Conference: Education and psychology challenges – teachers for the knowledge society - 2nd edition epc – tks 2013, 211-216, ISSN 1877-0428,� 11. Kadelburg, Z., Mladenović, P.: Savezna takmičenja iz matematike, DMS, Beograd, 1990 12. Mateljević, M.: Kompleksne funkcije 1 & 2, DMS, Beograd, 2006 13. Mitrinović, D. S.: Kompleksna analiza, zbornik zadataka i problema, Naučna knji­ga, Beograd, 1972 14. Mitrović, M.; Ognjanović, S.; Veljković, M.; Petković, Lj.; Lazarević, N.: Geometrija za I razred Matematičke gimnazija, Krug, Beograd, 1998 15. Mladenović, P.; Ognjanović, S.: Pripremni zadaci za matematička takmičenja za učenike srednjih škola, DMS, Beograd, 1991

236

16. Pavković, B.; Veljan, D.: Elementarna matematika I, Tehnička knjiga, Zagreb, 1992 17. Spiegel, A. M.: Theory and problems of complex variabbles, McGraw-Hill Book Co., Singapore, 1981 18. Бойваленков, П.; Николов, Н.: Национални олимпиади по математика 2008, УНИМАТ СМБ, София, 2008 19. Гроздев, С., Ненков, В. Геометрични места при движение на вписан четириъгълник. Математика и математическо образование, 2013, 42, 373-382, ISSN����������� 1313-3330. 20. Гроздев, С., Ненков, В. Допирателни окръжности, породени от инциденти точка и права, 2012, 2, 151 – 160,���������������� ��������������� ISSN����������� 1310-2230�. 21. Гроздев, С., Ненков, В. Една зависимост, породена от конични сечения. Мате­ матика и математическо образование, 2008, 37, 312 – 319, ISSN����������� ��������������� 1313-3330�. 22. Гроздев, С., Ненков, В.: Около ортоцентъра в равнината и пространството. София, Архимед 2, 2012, ISBN 978-954-779-145-9. 23. Гроздев, С., Ненков, В.: Ортоцентър на вписан четириъгълник, Математика плюс, 2011, 4, 63 – 69, ISSN 0861-8321. 24. Јанковић, З.; Каделбург, З.; Младеновић, П.: Међународне и балканске матема­ тичке олимпијаде 1984-1995, ДМС, Београд, 1995 25. Карамата, Ј.: Комплексан број, са применом на елементарну геоме­трију, Научна књига, Београд, 1950 26. Кендеров, П., Табов, Й.: Бьлгарски олимпиади по математика, Народ­на просвета, София, 1990 27. Кртинић, Ђ.: Математичке олимпијаде средњошколаца 2007-2012 године, ДМ Србије, 2012 28. Мадески, Ж.; Самарџиски, А.; Целакоски, Н.: Збирка задачи по геометрија, Просветно дело, Скопје, 1981 29. Малчески, Р., Аневска, К.: Хомотетија, Сигма, 2014, Скопје 30. Малчески, Р.: Основи на математичка анализа, Ун. Св. Кирил и Методиј, Скопје, 2001 31. Малчески, Р.: Теорема на Менелај, Сигма 43, Скопје, 1999 32. Малчески, Р.; Малчески, А.: Пресликувања во рамнина преку комплекс­ни броеви I, Сигма 49, Скопје, 2000 33. Малчески, Р.; Малчески, А.: Пресликувања во рамнина преку комплекс­ни броеви II, Сигма 50, Скопје, 2000

237

34. Малчески, Р.; Малчески, А.: Пресликувања во рамнина преку комплекс­ни броеви III, Сигма 51, Скопје, 2001 35. Малчески, Р.; Малчески, А.: Пресликувања во рамнина преку комплекс­ни броеви IV, Сигма 52, Скопје, 2001 36. Моденов, П. Ц.: Задачи по геоме­трии, Наука, Москва, 1979 37. Ненков, В.: Върху една задача и нейното пълно решение, Математика и информатика, 1998, 3, 70 – 72, ISSN 1310-2230. 38. Ненков, В.: Комплексни числа и педални окръжности, Математика и информатика, 1997, 5-6, 37 – 41, ISSN 1310-2230. 39. Ненков, В.: Някои геометрични места в равнината на триъгълника, Математика плюс, 2005, 1, 53 – 59, ISSN 0861-8321. 40. Ненков, В.: Четири Ойлерови прави в равнината на триъгълник, Математика и информатика, 2005, 2, 68 – 80, ISSN 1310-2230 41. Рудин, У.: Основи на математическая анализ, Наука и изкуство, София, 1973 42. Самарџиски, А.: Хомотетија, ин­вер­зија и задачите на Аполониј, ПМФ, Скопје, 1988 43. Сидоров, Ю. В.; Федорюк, М. В.; Шабунин, М. И.: Лекции по тории функций комплексного переменного, Наука, Москва, 1989 44. Тонов, И. К.; Сидеров, П. Н.: Прило­жение на комплексните числа в геометрията, Наука, София, 1981 45. Тренчевски, К.; Урумов, В.: Меѓународни олимпијади по математика, Природноматематички факултет, Скопје, 2000 46. Шабат, Б. В.: Введение в комплексный анализ, часть I, Наука, Москва, 1976

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GEOMETRY OF COMPLEX NUMBERS Risto Malcheski, Sava Grozdev, Katerina Anevska Bulgarian. First edition in english Printed by Alliance print – Sofia

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