1. Geometry in the 21 st Century Dynamic and Symbolic Douglas B. Meade Department of Mathematics University of South Carolina E-mail: meade@math.sc.edu meade@math.sc.edu Homepage: www.math.sc.edu/~meade/ www.math.sc.edu/~meade/

2. 4 February 2006 SC Junior Academy of Science Winter Conference (SVHS)2 Outline Euclidean Geometry Euclidean Geometry Euclidean Geometry Euclidean Geometry Conic Sections Conic Sections Conic Sections Conic Sections Shrinking Circles Shrinking Circles Shrinking Circles Shrinking Circles

3. 4 February 2006 SC Junior Academy of Science Winter Conference (SVHS)3 Euclidean Geometry Napoleon’s Theorem Napoleon’s Theorem  Given any  ABC  Construct an equilateral triangle on each side of  ABC  Construct  GIK joining the incenters of the three equilateral triangles  What can be said about  GIK? What can be said about  GIK? What can be said about  GIK?

4. 4 February 2006 SC Junior Academy of Science Winter Conference (SVHS)4 Conic Sections Ellipse Ellipse Ellipse Hyperbola Hyperbola Hyperbola Parabola Parabola Parabola

5. 4 February 2006 SC Junior Academy of Science Winter Conference (SVHS)5 Ellipse Definition An ellipse is a curve that is the locus of all points in the plane the sum of whose distances from two fixed points (the foci) is a given positive constant. Definition An ellipse is a curve that is the locus of all points in the plane the sum of whose distances from two fixed points (the foci) is a given positive constant.

6. 4 February 2006 SC Junior Academy of Science Winter Conference (SVHS)6 Hyperbola Definition A hyperbola is a curve defined as the locus of all points in the plane the difference of whose distances from two fixed points (the foci) a given positive constant. Definition A hyperbola is a curve defined as the locus of all points in the plane the difference of whose distances from two fixed points (the foci) a given positive constant.

7. 4 February 2006 SC Junior Academy of Science Winter Conference (SVHS)7 Parabola Definition A parabola is the set of all points in the plane equidistant from a given line (the conic section directrix) and a given point not on the line (the focus). Definition A parabola is the set of all points in the plane equidistant from a given line (the conic section directrix) and a given point not on the line (the focus).

8. 4 February 2006 SC Junior Academy of Science Winter Conference (SVHS)8 Shrinking Circles Construction   C * – unit circle centered at (1,0)   C r – circle with radius r centered at origin   P – a (0, r )   Q – intersection of C * and C r in first quadrant   R – intersection of the line PQ and the x -axis

9. 4 February 2006 SC Junior Academy of Science Winter Conference (SVHS)9 Shrinking Circles Question What happens to R as C r shrinks i.e., in the limit as r decreases towards 0?

10. 4 February 2006 SC Junior Academy of Science Winter Conference (SVHS)10 Generalization Suppose is replaced with an ellipse that passes through the origin: Suppose C * is replaced with an ellipse that passes through the origin:  What happens to R ? as shrinks?  What happens to R ? as C r shrinks?  Does it matter if an axis of the ellipse is on the x -axis. What if is replaced with a general curve that passes through the origin? What if C * is replaced with a general curve that passes through the origin?

11. 4 February 2006 SC Junior Academy of Science Winter Conference (SVHS)11 Generalization 3-D Construction   S * – unit sphere centered at (1,0,0)   S r – sphere with radius r centered at origin   P – a (0, 0, r )   C – intersection (curve) of S * and S r   R – intersection (curve) of cone with vertex P and cross-section C with the xy -plane

12. 4 February 2006 SC Junior Academy of Science Winter Conference (SVHS)12 Conclusions Geometry Geometry  is more than pencil and paper constructions  is more than two-column proofs  can be dynamic  is relevant to calculus, and beyond

13. 4 February 2006 SC Junior Academy of Science Winter Conference (SVHS)13 Relevant Software Geometry Expressions (beta) http://www.geometryexpressions.com/ Geometry Expressions (beta) http://www.geometryexpressions.com/ http://www.geometryexpressions.com/ Geometer’s Sketchpad http://www.keypress.com/sketchpad/ Geometer’s Sketchpad http://www.keypress.com/sketchpad/ http://www.keypress.com/sketchpad/ Cabri3D http://www.cabri.com/ Cabri3D http://www.cabri.com/ http://www.cabri.com/ Maple http://www.maplesoft.com/ Maple http://www.maplesoft.com/ http://www.maplesoft.com/