Topics Covered (tentative!):
The Basics:WRITING UP HOMEWORK SOLUTIONS: You are expected to show all essential steps in solving a problem. Draw appropriate figures. If in doubt, PUT IT IN! Make your solution flow. Ask yourself the question: "If I were the reader, how would I score this problem?'' Numerical answers alone, single words "yes" or "no", etc., are NEVER, EVER acceptable as complete answers. One of the skills you are expected to learn in this class is that of communicating mathematics clearly and completely in writing. Writing up homework exercises is your opportunity to hone these skills. Your grade depends on it!
Congruent triangles, angles and parallel lines, parallelograms area,
circles and arcs, polygons in circles, similarity.
Triangles:
The Circumcircle, the centroid, the Euler line, orthocenter, and nine-point
circle, computations, the incircle, excribed circles, Morley's Theorem,
optimization in triangles.
Circles and Lines:
Simson lines, the Butterfly Theorem, cross ratios, the radical axis.
Ceva's Theorem and Its Relatives:
Ceva's Theorem, interior and exterior Cevians, angles, Menelaus' Theorem.
Vector Methods of Proof:
Vectors, geometry, dot products, checkerboards, trigonometry, linear
operators.
Geometric Constructions:
The rules, triangles, tangents, hard problems, constructible numbers,
changing the rules.
Non-Euclidean Geometry:
Euclid's 5-th postulate. Hyperbolic: Saccheri quadrilateral, the angle sum
of a triangle, models. Elliptic: the angle sum of a triangle, models.
Euclid In Higher Dimensions:
The Platonic solids, distance, volume, Kepler's Problem.
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