Ion Pătrașcu Florentin Smarandache THE GEOMETRY OF THE ORTHOLOGICAL TRIANGLES Peer Reviewers: Prof. dr. Bencze Mihály Department of Mathematics, “Áprily Lajos” Highschool, Brașov, Romania Prof. univ. dr. Temistocle Bîrsan “Gh. Asachi” Technical University, Iași, Romania Lect. univ. dr. Claudiu-Ionuț Popîrlan Department of Computer Science, Faculty of Sciences, University of Craiova, Romania Ion Pătrașcu Florentin Smarandache THE GEOMETRY OF THE ORTHOLOGICAL TRIANGLES Pons Editions Brussels, 2020 To my grandchildren LUCAS and EVA-MARIE, with all my love – ION PĂTRAȘCU ISBN 978-1-59973-653-2 DTP: Alexandre Lukacs Translation from Romanian by Oana Dumitru Pons Editions Quai du Batelage, 5 1000 - Brussels TABLE OF CONTENTS TABLE OF CONTENTS FOREWORD ................................................................................................ 21 AUTHORS’ NOTE ....................................................................................... 23 1 INTRODUCTION ..................................................................................... 25 1.1 ORTHOLOGICAL TRIANGLES: DEFINITION .............................................. 25 DEFINITION 1 ................................................................................................... 26 OBSERVATION 1 .............................................................................................. 26 EXERCISE 1...................................................................................................... 26 1.2. CHARACTERIZATION OF THE ORTHOLOGY RELATION .......................... 26 THEOREM 1 (L. CARNOT, 1803) ...................................................................... 26 PROOF ............................................................................................................. 27 OBSERVATION 2 .............................................................................................. 28 LEMMA 1 ......................................................................................................... 28 EXERCISE 2...................................................................................................... 28 THEOREM 2 ..................................................................................................... 29 PROOF ............................................................................................................. 29 THEOREM 3 ..................................................................................................... 30 PROOF ............................................................................................................. 30 OBSERVATION 3 .............................................................................................. 31 1.3. THE ORTHOLOGICAL TRIANGLES THEOREM ......................................... 31 THEOREM 4 (J. STEINER, 1828 – THEOREM OF ORTHOLOGICAL TRIANGLES) ... 31 PROOF 1 .......................................................................................................... 31 PROOF 2 .......................................................................................................... 31 PROOF 3 .......................................................................................................... 32 PROOF 4 .......................................................................................................... 33 PROOF 5 .......................................................................................................... 34 REMARK 1 ....................................................................................................... 35 REMARK 2 ....................................................................................................... 35 PROOF 6 (ANALYTICAL) ................................................................................... 35 DEFINITION 2 ................................................................................................... 37 PROBLEM 1 ...................................................................................................... 37 PROBLEM 2 ...................................................................................................... 37 Ion Pătrașcu, Florentin Smarandache 5 THE GEOMETRY OF THE ORTHOLOGICAL TRIANGLES TABLE OF CONTENTS 2 ORTHOLOGICAL REMARKABLE TRIANGLES ............................. 39 2.1 A TRIANGLE AND ITS COMPLEMENTARY TRIANGLE ............................... 39 DEFINITION 3 ................................................................................................... 39 PROPOSITION 1 ................................................................................................ 39 PROOF ............................................................................................................. 39 OBSERVATION 4 .............................................................................................. 40 PROBLEM 3 ...................................................................................................... 40 2.2 A TRIANGLE AND ITS ANTI-COMPLEMENTARY TRIANGLE ...................... 41 DEFINITION 4 ................................................................................................... 41 PROPOSITION 2 ................................................................................................ 41 PROOF ............................................................................................................. 41 OBSERVATION 5 .............................................................................................. 42 2.3 A TRIANGLE AND ITS ORTHIC TRIANGLE ................................................. 42 DEFINITION 5 ................................................................................................... 42 OBSERVATION 6 .............................................................................................. 42 DEFINITION 6 ................................................................................................... 43 OBSERVATION 7 .............................................................................................. 43 PROPOSITION 3 ................................................................................................ 43 PROOF ............................................................................................................. 44 REMARK 3 ....................................................................................................... 44 PROPOSITION 4 ................................................................................................ 44 PROOF ............................................................................................................. 44 PROPOSITION 5 ................................................................................................ 45 PROOF ............................................................................................................. 45 REMARK 4 ....................................................................................................... 46 PROPOSITION 6 ................................................................................................ 46 PROOF ............................................................................................................. 46 REMARK 5 ....................................................................................................... 47 OBSERVATION 8 .............................................................................................. 47 2.4 THE MEDIAN TRIANGLE AND THE ORTHIC TRIANGLE ............................. 47 THEOREM 5 ..................................................................................................... 47 PROOF ............................................................................................................. 47 OBSERVATION 9 .............................................................................................. 48 PROPOSITION 7 ................................................................................................ 49 PROPOSITION 8 ................................................................................................ 49 PROPOSITION 9 ................................................................................................ 49 PROOF ............................................................................................................. 49 OBSERVATION 10 ............................................................................................ 49 6 Ion Pătrașcu, Florentin Smarandache THE GEOMETRY OF THE ORTHOLOGICAL TRIANGLES TABLE OF CONTENTS PROBLEM 4 ...................................................................................................... 49 PROBLEM 5 ...................................................................................................... 50 2.5 A TRIANGLE AND ITS CONTACT TRIANGLE .............................................. 50 DEFINITION 7 ................................................................................................... 50 OBSERVATION 11 ............................................................................................ 50 PROPOSITION 10 .............................................................................................. 50 PROOF ............................................................................................................. 51 OBSERVATION 12 ............................................................................................ 51 PROPOSITION 11 .............................................................................................. 51 PROOF 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