THE ELEMENTS OF EUCLID
CONTENTS PART I EUCLIDEAN GEOMETRY OF THE PLANE 1 THE BASIC NOTIONS 3 1 1 UNDEFINED TERMS AXIOMS 3 1 2 LINE AND LINE SEGMENT 6 1 3 LENGTH DISTANCE 9 1 4 ANGLES 12 1 5 ANGLE KINDS 16 1 6 TRIANGLES 19 1 7 CONGRUENCE THE EQUALITY OF SHAPES 23 1 8 ISOSCELES AND RIGHT TRIANGLE 25 1 9 TRIANGLE CONGRUENCE CRITERIA 27 1 10 TRIANGLE S SIDES AND ANGLES RELATIONS 33 1 11 THE TRIANGLE INEQUALITY 40 1 12 THE ORTHOGONAL TO A LINE 42 1 13 THE PARALLEL FROM A POINT 45 1 14 THE SUM OF TRIANGLE S ANGLES 48 1 15 THE AXIOM OF PARALLELS 52 1 16 SYMMETRIES 61 1 17 RATIOS HARMONIC QUADRUPLES 66 1 18 COMMENTS AND EXERCISES FOR THE CHAPTER 72 REFERENCES 77 2 CIRCLE AND POLYGONS 79 2 1 THE CIRCLE THE DIAMETER THE CHORD 79 2 2 CIRCLE AND LINE 84 2 3 TWO CIRCLES 88 2 4 CONSTRUCTIONS USING RULER AND COMPASS 91 2 5 PARALLELOGRAMS 101 2 6 QUADRILATERALS 105 2 7 THE MIDDLES OF SIDES 110 2 8 THE TRIANGLE S MEDIANS 114 2 9 THE RECTANGLE AND THE SQUARE 118 2 10 OTHER KINDS OF QUADRILATERALS 123 2 11 POLYGONS REGULAR POLYGONS 127 2 12 ARCS CENTRAL ANGLES 135 2 13 INSCRIBED ANGLES 139 2 14 INSCRIPTIBLE OR CYCLIC QUADRILATERALS 151 2 15 CIRCUMSCRIBED QUADRILATERALS 158 2 16 GEOMETRIC LOCI 160 2 17 COMMENTS AND EXERCISES FOR THE CHAPTER 164 REFERENCES 180 3 AREAS THALES PYTHAGORAS PAPPUS 181 3 1 AREA OF POLYGONS 181 3 2 THE AREA OF THE RECTANGLE 183 3 3 AREA OF PARALLELOGRAM AND TRIANGLE 187 3 4 PYTHAGORAS AND PAPPUS 196 3 5 SIMILAR RIGHT TRIANGLES 205 3 6 THE TRIGONOMETRIC FUNCTIONS 213 3 7 THE THEOREM OF THALES 221 3 8 PENCILS OF LINES 227 3 9 SIMILAR TRIANGLES 237 3 10 SIMILAR POLYGONS 246 3 11 TRIANGLE S SINE AND COSINE RULES 258 3 12 STEWART MEDIANS BISECTORS ALTITUDES 268 3 13 ANTIPARALLELS SYMMEDIANS 277 3 14 COMMENTS AND EXERCISES FOR THE CHAPTER 284 REFERENCES 309 4 THE POWER OF THE CIRCLE 311 4 1 POWER WITH RESPECT TO A CIRCLE 311 4 2 GOLDEN SECTION AND REGULAR PENTAGON 316 4 3 RADICAL AXIS RADICAL CENTER 321 4 4 APOLLONIAN CIRCLES 327 4 5 CIRCLE PENCILS 333 4 6 ORTHOGONAL CIRCLES AND PENCILS 344 4 7 SIMILARITY CENTERS OF TWO CIRCLES 352 4 8 INVERSION 360 4 9 POLAR AND POLE 372 4 10 COMMENTS AND EXERCISES FOR THE CHAPTER 379 REFERENCES 412 5 FROM THE CLASSICAL THEOREMS 413 5 1 ESCRIBED CIRCLES AND EXCENTERS 413 5 2 HERON S FORMULA 419 5 3 EULER S CIRCLE 422 5 4 FEUERBACH S THEOREM 427 5 5 EULER S THEOREM 429 5 6 TANGENT CIRCLES OF APOLLONIUS 436 5 7 THEOREMS OF PTOLEMY AND BRAHMAGUPTA 443 5 8 SIMSON S AND STEINER S LINES 452 5 9 MIQUEL POINT PEDAL TRIANGLE 457 5 10 ARBELOS 466 5 11 SANGAKU 472 5 12 FERMAT S AND FAGNANO S THEOREMS 477 5 13 MORLEY S THEOREM 482 5 14 SIGNED RATIO AND DISTANCE 485 5 15 CROSS RATIO HARMONIC PENCILS 493 5 16 THEOREMS OF MENELAUS AND CEVA 501 5 17 THE COMPLETE QUADRILATERAL 513 5 18 DESARGUES THEOREM 521 5 19 PAPPUS THEOREM 526 5 20 PASCAL S AND BRIANCHON S THEOREMS 530 5 21 CASTILLON S PROBLEM HOMOGRAPHIC RELATIONS 540 5 22 MALFATTI S PROBLEM 549 5 23 CALABI S TRIANGLE 553 5 24 COMMENTS AND EXERCISES FOR THE CHAPTER 557 REFERENCES 587 PART I EUCLIDEAN GEOMETRY OF THE PLANE