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The Stuyvesant High School Online Course Guide |
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The Stuyvesant High School Online Course Guide |
Who should take this course?
This is a new course intended for juniors and seniors who have completed a two-year sequence in introductory geometry and trigonometry, and wish to explore the rich subject of plane geometry at much greater depth. Students should demonstrate the talent and ability to excel in mathematics at a collegiate level, and should also show a strong appreciation for the intrinsic beauty of geometry. If you are prone to amazement by unexpected relationships between concepts, then you may well enjoy this course.
What is Advanced Euclidean Geometry?
The triangle is perhaps the simplest figure studied in elementary geometry, having been well understood since the time of the Greek geometers. However, some of the most fascinating properties of triangles escaped the Greeks, and were not discovered until the Renaissance in Europe. In fact, many of the ideas we will study have not been known until recent years. For example, a classic theorem states that for any triangle, the midpoints of the sides, the feet of the altitudes, and the midpoints of the orthocentral segments all lie on a single circle, whose radius is exactly half the radius of the circumscribed circle. Moreover, this circle is always internally tangent to the inscribed circle of the triangle. Examples like these bring out the enormous depth of the geometry of triangles, and the extent to which completely unexpected relationships exist between seemingly the simplest concepts. Modern triangle geometry has been described as the field in mathematics with the highest number of miracles per page.What will be expected of me?
Evaluation will be based largely on written work. There will be five in-term examinations and a final examination. In addition, there will be biweekly problem sets that will include some classic challenge problems and also some longer investigation problems. Students will be required to write a research paper.
How will this course be taught?
The course will be taught in the manner and at the level of a typical college course for mathematics majors. Proof will be emphasized over computation. Students must be comfortable with both the meaning and the methodology of proof. Class time will be primarily reserved for lectures on the proofs of the classical and modern theorems of triangle geometry. A significant amount of work outside the classroom will be expected of students.
Details and Pre-requisites
Students should have completed MQ6 (Trigonometry) with a minimum grade of 88% and should also have an overall math average of at least 88%.
