In the figure below, prove that GH divides AB harmonically by applying the Ceva and Menelaus theorems to triangle ABE. [Be able to state the theorems and definitions that you use.]
Be able to state the connection between harmonic division and inversion and also the connection with cross-ratio.
State Desargues theorem and explain why the 3-dimensional version of the theorem is true (in terms of intersections of planes).
Given a circle and a point G, explain how to construct the polar of G from two secants through P intersecting the circle in AB and CD (as in the figure above). Which line is the polar and why. Then explain why the polar can be defined for any conic using this construction. Why does this not work to define inversion in a conic?
Review the questions in the Final Exam Winter 2000.
