(Classroom Resource Materials)
Gerard A. Venema
Mathematical Association of America | 2013 | 146 páginas | rar - pdf | 820 kb
link (password: matav)
This book provides an inquiry-based introduction to advanced Euclidean geometry. It utilizes dynamic geometry software, specifically GeoGebra, to explore the statements and proofs of many of the most interesting theorems in the subject. Topics covered include triangle centers, inscribed, circumscribed, and escribed circles, medial and orthic triangles, the nine-point circle, duality, and the theorems of Ceva and Menelaus, as well as numerous applications of those theorems. The final chapter explores constructions in the Poincaré disk model for hyperbolic geometry.
The book can be used either as a computer laboratory manual to supplement an undergraduate course in geometry or as a stand-alone introduction to advanced topics in Euclidean geometry. The text consists almost entirely of exercises (with hints) that guide students as they discover the geometric relationships for themselves. First the ideas are explored at the computer and then those ideas are assembled into a proof of the result under investigation. The goals are for the reader to experience the joy of discovering geometric relationships, to develop a deeper understanding of geometry, and to encourage an appreciation for the beauty of Euclidean geometry.
Contents
0 A Quick Review of Elementary Euclidean Geometry 1
0.1 Measurement and congruence. . 1
0.2 Angle addition . . 2
0.3 Triangles and triangle congruence conditions . . 3
0.4 Separation and continuity. . 4
0.5 The exterior angle theorem . . 5
0.6 Perpendicular lines and parallel lines . . 5
0.7 The Pythagorean theorem. . 7
0.8 Similar triangles . . . 8
0.9 Quadrilaterals . . 9
0.10 Circles and inscribed angles . . 10
0.11 Area . . 11
1 The Elements of GeoGebra 13
1.1 Getting started: the GeoGebra toolbar. . 13
1.2 Simple constructions and the drag test . . 16
1.3 Measurement and calculation . . 18
1.4 Enhancing the sketch .. . 20
2 The Classical Triangle Centers 23
2.1 Concurrent lines . . 23
2.2 Medians and the centroid . . 24
2.3 Altitudes and the orthocenter .. 25
2.4 Perpendicular bisectors and the circumcenter . . 26
2.5 The Euler line . . 27
3 Advanced Techniques in GeoGebra 31
3.1 User-defined tools . . 31
3.2 Check boxes . . 33
3.3 The Pythagorean theorem revisited . . 34
4 Circumscribed, Inscribed, and Escribed Circles 39
4.1 The circumscribed circle and the circumcenter . . 39
4.2 The inscribed circle and the incenter . . 41
4.3 The escribed circles and the excenters . . 42
4.4 The Gergonne point and the Nagel point . . 43
4.5 Heron’s formula . . 44
5 The Medial and Orthic Triangles 47
5.1 The medial triangle . . 47
5.2 The orthic triangle . . 48
5.3 Cevian triangles . . 50
5.4 Pedal triangles . . 51
6 Quadrilaterals 53
6.1 Basic definitions . . 53
6.2 Convex and crossed quadrilaterals. . 54
6.3 Cyclic quadrilaterals . . 55
6.4 Diagonals . . 56
7 The Nine-Point Circle 57
7.1 The nine-point circle . . . 57
7.2 The nine-point center . . 59
7.3 Feuerbach’s theorem . . 60
8 Ceva’s Theorem 63
8.1 Exploring Ceva’s theorem . . 63
8.2 Sensed ratios and ideal points . . 65
8.3 The standard form of Ceva’s theorem . . 68
8.4 The trigonometric form of Ceva’s theorem . . 71
8.5 The concurrence theorems . . 72
8.6 Isotomic and isogonal conjugates and the symmedian point . . 73
9 The Theorem of Menelaus 77
9.1 Duality . . 77
9.2 The theorem of Menelaus . . 78
10 Circles and Lines 81
10.1 The power of a point . . 81
10.2 The radical axis . . 83
10.3 The radical center . . 84
11 Applications of the Theorem of Menelaus 85
11.1 Tangent lines and angle bisectors . . . 85
11.2 Desargues’ theorem . . 86
11.3 Pascal’s mystic hexagram . . 88
11.4 Brianchon’s theorem . . 90
11.5 Pappus’s theorem . . 91
11.6 Simson’s theorem. . 93
11.7 Ptolemy’s theorem . . 96
11.8 The butterfly theorem . . 97
12 Additional Topics in Triangle Geometry 99
12.1 Napoleon’s theorem and the Napoleon point . . 99
12.2 The Torricelli point . . 100
12.3 van Aubel’s theorem. . 100
12.4 Miquel’s theorem and Miquel points . . 101
12.5 The Fermat point . . 101
12.6 Morley’s theorem . . 102
13 Inversions in Circles 105
13.1 Inverting points . . 105
13.2 Inverting circles and lines . . 107
13.3 Othogonality . . 108
13.4 Angles and distances. . 110
14 The Poincar´e Disk 111
14.1 The Poincar´e disk model for hyperbolic geometry . . 111
14.2 The hyperbolic straightedge . . 113
14.3 Common perpendiculars . . 114
14.4 The hyperbolic compass. . 116
14.5 Other hyperbolic tools . . 117
14.6 Triangle centers in hyperbolic geometry . . 118
References 121
Index 123
