ANALYTICAL SOLID GEOMETRY
CONTENTS CHAPTER 1 SIMILAR TRIANGLES CHAPTER 2 INSCRIBED ANGLES CHAPTER 3 CIRCLES CHAPTER 4 AREA CHAPTER 5 TRIANGLES CHAPTER 6 POLYGONS CHAPTER 7 LOCI CHAPTER 8 CONSTRUCTIONS CHAPTER 9 GEOMETRIC INEQUALITIES CHAPTER 10 INEQUALITIES BETWEEN THE ELEMENTS OF A TRIANGLE CHAPTER 11 PROBLEMS ON MAXIMUM AND MINIMUM CHAPTER 12 CALCULATIONS AND METRIC RELATIONS CHAPTER 13 VECTORS CHAPTER 14 THE CENTER OF MASS CHAPTER 15 PARALLEL TRANSLATIONS CHAPTER 16 CENTRAL SYMMETRY CHAPTER 17 THE SYMMETRY THROUGH A LINE CHAPTER 18 ROTATIONS CHAPTER 19 HOMOTHETY AND ROTATIONAL HOMOTHETY CHAPTER 20 THE PRINCIPLE OF AN EXTREMAL ELEMENT CHAPTER 21 DIRICHLET’S PRINCIPLE CHAPTER 22 CONVEX AND NONCONVEX POLYGONS CHAPTER 23 DIVISIBILITY, INVARIANTS, COLORINGS CHAPTER 24 INTEGER LATTICES CHAPTER 25 CUTTINGS CHAPTER 26 SYSTEMS OF POINTS AND SEGMENTS EXAMPLES AND COUNTEREXAMPLES CHAPTER 27 INDUCTION AND COMBINATORICS CHAPTER 28 INVERSION CHAPTER 29 AFFINE TRANSFORMATIONS CHAPTER 30 PROJECTIVE TRANSFORMATIONS CHAPTER 1 SIMILAR TRIANGLES BACKGROUND 1) TRIANGLE ABC IS SAID TO BE SIMILAR TO TRIANGLE A1B1C1 (WE WRITE ?ABC ??A1B1C1) IF AND ONLY IF ONE OF THE FOLLOWING EQUIVALENT CONDITIONS IS SATIS?ED: A) AB : BC : CA = A1B1 : B1C1 : C1A1; B) AB : BC = A1B1 : B1C1 AND ?ABC = ?A1B1C1; C) ?ABC = ?A1B1C1 AND ?BAC = ?B1A1C1 2) TRIANGLES AB1C1 AND AB2C2 CUT O?FROM AN ANGLE WITH VERTEX A BY PARALLEL LINES ARE SIMILAR AND AB1 : AB2 = AC1 : AC2 (HERE POINTS B1 AND B2 LIE ON ONE LEG OF THE ANGLE AND C1 AND C2 ON THE OTHER LEG) 3) A MIDLINE OF A TRIANGLE IS THE LINE CONNECTING THE MIDPOINTS OF TWO OF THE TRIANGLE’S SIDES THE MIDLINE IS PARALLEL TO THE THIRD SIDE AND ITS LENGTH IS EQUAL TO A HALF LENGTH OF THE THIRD SIDE THE MIDLINE OF A TRAPEZOID IS THE LINE CONNECTING THE MIDPOINTS OF THE TRAPEZOID’S SIDES THIS LINE IS PARALLEL TO THE BASES OF THE TRAPEZOID AND ITS LENGTH IS EQUAL TO THE HALFSUM OF THEIR LENGTHS 4) THE RATIO OF THE AREAS OF SIMILAR TRIANGLES IS EQUAL TO THE SQUARE OF THE SIMILARITY COE?CIENT, I E , TO THE SQUARED RATIO OF THE LENGTHS OF RESPECTIVE SIDES THIS FOLLOWS, FOR EXAMPLE, FROM THE FORMULA SABC = 2AB AC SIN ?A 5) POLYGONS A1A2 AN AND B1B2 BN ARE CALLED SIMILAR IF A1A2 : A2A3 : : ANA1 = B1B2 : B2B3 : : BNB1 AND THE ANGLES AT THE VERTICES A1, , AN ARE EQUAL TO THE ANGLES AT THE VERTICES B1, , BN, RESPECTIVELY THE RATIO OF THE RESPECTIVE DIAGONALS OF SIMILAR POLYGONS IS EQUAL TO THE SIMILARITY COE?- CIENT FOR THE CIRCUMSCRIBED SIMILAR POLYGONS, THE RATIO OF THE RADII OF THE INSCRIBED CIRCLES IS ALSO EQUAL TO THE SIMILARITY COE?CIENT INTRODUCTORY PROBLEMS 1 CONSIDER HEIGHTS AA1 AND BB1 IN ACUTE TRIANGLE ABC PROVE THAT A1C BC = B1C AC 2 CONSIDER HEIGHT CH IN RIGHT TRIANGLE ABC WITH RIGHT ANGLE ?C PROVE THAT AC2 = AB AH AND CH2 = AH BH 3 PROVE THAT THE MEDIANS OF A TRIANGLE MEET AT ONE POINT AND THIS POINT DIVIDES EACH MEDIAN IN THE RATIO OF 2 : 1 COUNTING FROM THE VERTEX 4 ON SIDE BC OF ?ABC POINT A1 IS TAKEN SO THAT BA1 : A1C = 2 : 1 WHAT IS THE RATIO IN WHICH MEDIAN CC1 DIVIDES SEGMENT AA1? 5 SQUARE PQRS IS INSCRIBED INTO ?ABC SO THAT VERTICES P AND Q LIE ON SIDES AB AND AC AND VERTICES R AND S LIE ON BC EXPRESS THE LENGTH OF THE SQUARE’S SIDE THROUGH A AND HA §1 LINE SEGMENTS INTERCEPTED BY PARALLEL LINES 1 1 LET THE LENGTHS OF BASES AD AND BC OF TRAPEZOID ABCD BE A AND B (A > B)