Modern Pure Solid Geometry [1 ed.]

Table of contents :
PREFACE ............................................... v

CHAPTER I. PRELIMINARY ................................ 1
1. Introductory Propositions and Problems ........ 1
2. The Hyperbolic Group of Lines ................. 9
3. Harmonic Forms ................................ 12
4. Homothetic Figures ............................ 16
5. Perspective Tetrahedrons ...................... 21

CHAPTER II. THE TRIHEDRAL ANGLE ....................... 27
1. The Orthocentric Line ......................... 27
2. The Supplementary Trihedral Angle ............. 30
3. Isoclinal Lines and Planes .................... 32
4. Axes .......................................... 37
5. Centroidal Lines .............................. 40

CHAPTER III. THE SKEW QUADRILATERAL ................... 42

CHAPTER IV. THE TETRAHEDRON ........................... 48
1. The Centroid .................................. 48
a. Bimedians and Bialtitudes .................. 48
b. Medians .................................... 51
c. The Circumscribed Parallelepiped ........... 58
2. Altitudes ..................................... 61
a. Special Cases .............................. 61
b. The General Case ........................... 66
c. The Monge Point ............................ 68
3. The Bisecting Planes of the Dihedral Angles ... 71
4. The Spheres Touching the Four Faces of a
Tetrahedron ................................... 72
a. The Inscribed Sphere ....................... 72
b. The Existence and Distribution of the
Escribed Spheres .......................... 74
c. Relative Position of the Centers to
Each Other ................................ 78
d. The Points of Contact with the Faces ....... 80
e. Relations Between the Radii ................ 82
5. Volume of Tetrahedron ......................... 86
6. Special Tetrahedrons .......................... 91
a. The Trirectangular Tetrahedron ............. 91
b. The Isosceles Tetrahedron .................. 94
7. Miscellaneous Propositions .................... 102

CHAPTER V. TRANSVERSALS ............................... 111
1. The Skew Quadrilateral ........................ 111
2. The Tetrahedron ............................... 115

CHAPTER VI. THE OBLIQUE CONE WITH A CIRCULAR BASE...... 123

CHAPTER VII. SPHERES .................................. 133
1. Preliminaries ................................. 133
2. Inverse Points ................................ 135
3. Orthogonal Spheres ............................ 136
4. Poles and Polar Planes with Respect to a
Sphere ........................................ 140
5. The Imaginary Sphere .......................... 149
6. Centers, Axes, and Planes of Similitude ....... 151
7. The Power of a Point with Respect to a Sphere . 161
8. The Radical Plane of Two Spheres .............. 167
9. Coaxal Pencils of Spheres ..................... 177
10. Coaxal Nets of Spheres ........................ 191
11. Four Spheres .................................. 201

CHAPTER VIII. INVERSION ............................... 214

CHAPTER IX. RECENT GEOMETRY OF THE TETRAHEDRON ........ 230
A. THE GENERAL TETRAHEDRON .......................... 230
1. Harmonic Points and Planes .................... 230
a. Tetrahedral Poles and Polar Planes ......... 230
b. Desmic Systems of Tetrahedrons ............. 235
2. Isogonal Points ............................... 240
3. Antiparallel Sections ......................... 247
4. Spheres Related to the Tetrahedron ............ 250
a. Analogues of the Nine-Point Circle ......... 250
b. The Apollonian Spheres of the Tetrahedron... 252
B. SPECIAL TETRAHEDRONS ............................. 256
1. The Circumscriptible Tetrahedron .............. 256
2. The Orthocentric Tetrahedron .................. 261
3. The Isodynamic Tetrahedron .................... 276
a. The Lemoine Point and the Lemoine Plane .... 276
b. The Lemoine Tetrahedron .................... 282
c. The Newberg Spheres ........................ 286
4. The Isogonic Tetrahedron ...................... 290

BIBLIOGRAPHICAL NOTES ................................. 297

INDEX ................................................. 305