Geometry
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1 Theorems |
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1.1 On Ptolmey's Theorem |
3 |
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1.2 Menelaus and Ceva |
8 |
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1.3 Napoleon's Theorem |
19 |
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1.4 Morley's Trisection Theorem |
29 |
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1.5 Pappus' Theorem |
36 |
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1.6 Pascal's Mystic Hexagram |
41 |
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1.7 Euclid's Proposition III,20 and its Converse |
51 |
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1.8 Generalizing Pythagoras and Carnot |
59 |
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2 On Conics |
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2.1 Loci of Equi-angular Points |
65 |
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2.2 Parabola Through Four Points |
68 |
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2.3 Revisiting the Four-Point Parabola |
73 |
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2.4 Normals From A Point To An Ellipse |
82 |
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2.5 On the Ellipse |
85 |
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2.6 Quadrilateral In a Circle |
93 |
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2.7 Areas of Conic Quadrilaterals |
101 |
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2.8 Quadrilateral Duality |
108 |
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2.9 Intersecting Circles |
112 |
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3 Calculus and Geometry |
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3.1 Curvature |
121 |
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3.2 The Curvatures of Hypersurfaces |
133 |
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3.3 Net Area and Green’s Theorem |
141 |
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3.4 Mean Distance from Vertex to Interior of Plane Figures |
145 |
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3.5 Distances in Bounded Regions |
151 |
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3.6 Constant Headings and Rhumb Lines |
158 |
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3.7 Enveloping Circular Arcs |
160 |
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3.8 Rolling Spheres and Cones |
165 |
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4 Algebraic Aspects |
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4.1 Generating Functions for Point Set Distances |
170 |
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4.2 Isospectral Point Sets in Higher Dimensions |
177 |
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4.3 Piano Keys |
181 |
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4.4 Sphere Packing in Curved 3D Space |
183 |
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4.5 The Orbit of Triangles |
186 |
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4.6 Equation of a Triangle |
191 |
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4.7
Cyclic Quadrilaterals Revisited 4.8 Conformal Coordinate Systems |
196 202 |
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4.9 The Five Squarable Lunes |
205 |
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5 Problems and Puzzles |
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5.1 Acute Problem |
213 |
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5.2 Pent Up Ratios |
215 |
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5.3 Angular Angst |
220 |
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5.4 Adventitious Solutions |
234 |
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5.5 Equidistant Curves |
241 |
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5.6 Thinking Outside the Triangle |
250 |
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5.7 Zeno’s Mice and the Logarithmic Spiral |
251 |
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5.8 Cutting Self-Similar Pentagons |
254 |
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5.9 Equal Bisectors and Isosceles Triangles |
256 |
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5.10 Thomson’s Problem |
260 |
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6 History |
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6.1 The Prismoidal Formula |
271 |
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6.2 Are All Triangles Isosceles? |
279 |
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6.3 Apollonius' Tangency Problem |
281 |
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6.4 Archimedes on Spheres and Cylinders |
285 |
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6.5 Did Archimedes Know Gauss-Bonnet? |
294 |
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6.6 Heron's Formula and Brahmagupta's Generalization |
295 |
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6.7 Heron's Formula for Tetrahedra |
298 |
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6.8 Piero della Francesca's Tetrahedron Formula |
303 |
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6.9 From Euclid to Gregory |
307 |
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6.10 Carnot, Organizer of Transversals |
314 |
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