The final exam is cumulative. It covers the topics from the first and second midterms,
together with inversions and the first notions of non-Euclidean geometry (as seen in
the Poincare model of the Lobachevskian geometry.) I have reorganized the topics to
make them look more like the exam question themes. (The items below are not supposed to
describe the actual questions. Some topics may be missing from the exam; some questions
may combine two or more different items.) There will be about 6-8 questions on the test.
(1) Tests for congruent triangles, special lines (medians, altitudes, bisectors),
properties of isosceles triangles, right triangles; exterior angles;
sum of angles in a triangle
(2) Quadrilaterals; properties of parallelograms, rectangles, rhombi; tests for special
quadrilaterals
(3) the triangle inequality and its applications (in particular, finding broken lines
that satisfy certain conditions and have shortest possible length)
(4) Circles: central angles; inscribed angles and related statements and their inverses,
(sections 122-126); inscribed and circumscribed circles for a triangle; radius OA
is perpendicular to the tangent line at A; degree of a point with respect a given
circle (sections 194-196).
(5) Construction problems (those based on (1)-(4) above, as well as on method of similarity).
You will have to explain how to carry out the construction with compass and straightedge,
and to prove that it yields the required object. You should know the basic constructions in
sections 61-67; although will not be required to reproduce all of them in every step
of your construction.
(6) Similar triangles (note that the second exam glided over this topic, as we haven't
practiced much with similar triangles before the test. Later on, we saw a lot of
applications and examples; the final exam will contains a problem along those lines).
(7) Plane transformations: isometries, homothety and similarity mappings,
inversions. Properties of these transformations, their compositions, etc.
(8) First notions of the Lobachevskian geometry (points, lines, angles and isometries
in the Poincare model).