MATHEMATICS W82

Exploring Advanced Euclidean Geometry
with Geometer's Sketchpad
Calvin College, Interim 2006

Instructor: Gerard A. Venema

Phone: 616-526-6402

Email: venema@calvin.edu

Homepage: http://www.calvin.edu/~venema

Textbook and course software

Each participant in the course will need a copy of the course software, The Geometer's Sketchpad (abbreviated GSP), published by Key Curriculum Press. The student edition is recommended and it is available at the Calvin College bookstore for $34.95.

Course goals

1. To explore the amazing and beautiful theorems of advanced Euclidean geometry
2. To learn to make effective use of dynamic geometry software
3. To discover an appropriate balance between computer exploration and proof

Some useful links

Course outline

1. A quick review of elementary Euclidean geometry
   • the crossbar theorem
   • linear pairs and vertical pairs
   • triangle congruence conditions
   • angles and parallel lines
   • the Pythagorean theorem
   • similar triangles
   • quadrilaterals
   • circles and inscribed angles
   • area
2. The elements of GSP
   • the toolbox
   • parents and children
   • constructions
   • measurement and calculation
   • enhancing the sketch
3. The classical triangle centers
   • concurrent lines
   • the centroid
   • the orthocenter
   • the circumcenter
   • the Euler line
4. Advanced techniques in GSP
   • custom tools
   • action buttons
   • the Euler line revisited
   • the Pythagorean theorem
5. Circumscribed circles, inscribed circles, and escribed circles
   • the circumscribed circle and the circumcenter
   • the inscribed circle and the incenter
   • escribed circles and the excenters
   • the Gergonne point and the Nagel point
6. Medial and orthic triangles
   • the medial triangle
   • the anticomplementary triangle
   • the orthic triangle
   • Cevian triangles
   • pedal triangles
7. The nine-point circle
   • the nine-point circle
   • the nine-point center
   • Feuerbach's theorem
8. Ceva's Theorem
   • exploring Ceva's theorem
   • sensed ratios and ideal points
   • the standard form of Ceva's theorem
   • the trigonometric form of Ceva's theorem
   • the concurrence theorems
     • existence of the centroid
     • existence of the orthocenter
     • existence of the circumcenter
     • existence of the incenter
     • existence of the Gergonne point
     • existence of the excenters
     • existence of the Nagel point
   • isotomic and isogonal conjugates and the symmedian point
9. The theorem of Menelaus
   • duality
   • the standard form of the theorem of Menelaus
   • the trigonometric form of the theorem of Menelaus
   • applications of the theorem of Menelaus
     • tangent lines and angle bisectors
     • Desargues's theorem
     • Pascal's mystic hexagram
     • Brianchon's theorem
     • Pappus's theorem
     • Simson's theorem and Simson lines
     • Ptolemy's theorem
10. Circles and lines
    • power of a point
    • radical axis
    • radical center
    • proof of Brianchon's theorem
    • the butterfly theorem
11. More topics in triangle geometry
    • Napoleon's theorem and the Napoleon point
      • van Aubel's theorem
    • the Fermat point
    • Morley's theorem
    • Miquel's theorem and the Miquel point
12. Transformations