- Task:
- Learn a new theorem in elementary geometry and communicate it in written form.
- Minimum requirements:
- Each student will focus on a different geometric result.
- The written document will introduce/motivate, correctly state, and prove a theorem. It will also include at least one interesting example, construction, or special case illustrating the theorem. The article will be as self-contained as possible. The new document must be typed, be at most eight pages in length (with one inch margins and a 12pt font), and be available in PDF format.
- Assessment and deadlines:
- Project grades will be computed as follows:
due date element weight 2015-02-03 outline 10% 2015-03-10 rough draft 20% 2015-03-17 feedback 20% 2012-03-31 final paper 50%
- Advice:
- Here are some suggestions on how to present mathematics: Halmos on writing and Kleiman on writing.
- Comments:
- By design, this assignment is very open-ended. Students are strongly
encouraged to explore examples. Students are also encouraged to
formulate, test, and prove their own conjectures. Here are some questions
that you may want to consider:
- What was the historical context or original motivation for your theorem?
- Can you give more than one proof of your theorem?
- Does your theorem have any interesting specializations or important applications?
- Can your theorem be generalized?
- Is your theorem valid in all geometries?
- Potential topics:
- The following results are natural candidates:
- Art Gallery Problem—Grushenka Ramhota
- Beck's Theorem
- Brahmagupta's Formula—Emily Dyson
- Brianchon's Theorem
- British Flag Theorem—Yi Zhang
- Butterfly Theorem—Melanie Ng
- Carnot's Theorem
- Cayley–Bacharach Theorem
- Ceva's Theorem—Natasha Mosdossy
- De Bruijn–Erdős Theorem
- Descartes' Theorem—Janice Fu
- Desargues' Theorem—Somee Park
- Droz–Farny Line Theorem
- Equal Incircles Theorem
- Euclidean Plane Isometries—Alex Haberfellner
- Fagnano's Problem—Shane Baccas
- Happy Ending Problem—Ashna Raman
- Heptadecagon—Alyson Rudnitski
- Heron's Formula—Emma Zahorchak
- Intercept Theorem—Brittney Camus
- Japanese Theorem—Kaylie Deblois
- Jung's Theorem—Alexandra Haberfellner
- Lester's Theorem
- Miquel's Theorem—Mark Geraedts
- Monge's Theorem—Delaney Hewage
- Morley's Trisector Theorem—John Meenagh
- Musselman's Theorem
- Napoleon's Theorem—Thomas Henbest
- Nine-point conic
- Pascal's Theorem—Liisa MacNicholas
- Pick's Theorem—Joanne Ching
- Pitot Theorem—Drishya Benjamin
- Pizza Theorem—Michelle Gruszecki
- Pompeiu's Theorem
- Poncelet's Theorem
- Pons Asinorum—Emma Hill
- Problem of Apollonius—Kathleen Eng
- Ptolemy's Theorem—Hartley Fanson
- Quadrature of the parabola—Natalie Stilo
- Routh's Theorem—Brodie Macleod
- Seven Circles Theorem—Elise Toussaint
- Six Circles Theorem
- Steiner–Lehmus Theorem—Esme Tremblay
- Stewart's Theorem—Tyler Flynn
- Sylvester–Gallai Theorem—Catherine Kern
- Szemerédi–Trotter Theorem—Mark Sunarich
- Varignon's Theorem—Rachel Youngson
- Viviani's Theorem—Jamie Ng
- Wallace–Bolyai–Gerwien Theorem—Ryan Kavanagh
- Weitzenböck's Inequality—Thomas Whaley
- Oene Bottema, Topics in elementary geometry, Second edition, Springer, New York, 2008.
- H.S.M. Coxeter and S.L. Greitzer, Geometry revisited, New Mathematical Library 19, Random House Inc., New York, 1967.
- Robin Hartshorne, Geometry: Euclid and beyond, UTM, Springer-Verlag, New York, 2000.
- David Hilbert, The Foundations of Geometry, Gutenberg eBook, 2005.
- Alexander Ostermann and Gerhard Wanner, Geometry by Its History, Springer, New York, 2012.
