Study Problems for Final Exam (not including problems from polyhedra)
Harmonic division and Apollonian circles
- Definition of harmonic division - ABCD.
- Definition of an Apollonian circle of A and B (as a locus)
- Prove that an Apollonian circle is really a circle.
- Construct the Apollonian circle of A and B through a given point P.
- Explain the relation between Apollonian circles and orthogonal circles.
- Reason that an Apollonian circle of A and B inverts A to B.
- Relation between harmonic division, inversion and orthogonal circles.
- Images of Euclidean figures under inversion as in Ogilvy, Chapter 4.
Inversive Geometry
- Prove that the inversion in a circle c of a line m is a circle m' (with
one special case as exception).
- Given a figure with two non-intersecting circles c and d, construct a circle
m so that the inversion of c and d are concentric circles.
- Given a figure with two non-intersecting circles c and d, construct a circle
m so that the inversions of c and d are a line and a circle.
- Write the correct formula for the inversion of point P(x,y) in the circle
with center (0,0) and radius r.
DWEG model
Be able to carry out the constructions of the lab.
- Construct a DWEG rectangle ABCD given DWEG points A and B.
- Construct a DWEG circle with center A through B (really Apollonian circle)
- Given DWEG lines AB and AC and a DWEG point P, why do the DWEG reflections
of P across these DWEG lines lie on an Apollianian circle?
- Explain in the DWEG model, given a DWEG line AB and a DWEG point C not on
AB, why is there exactly one DWEG line CD through C parallel to AB.
Poincare model
Be able to carry out the constructions from labs, including the ones below.
- Given a P-line AB and a P-point C, construct the limiting (asymptotic) parallels
to AB through C.
- Given a P-line AB and a P-point C, construct the perpendicular P-line to
AB through C.
- Given a P-line AB and a P-line CD, construct the P-line that is perpendicular
to both the given P-lines.
- Given P-points A and B, construct the P-line that reflects A to B. (Special
case: what is the construction if B is the Euclidean center of the boundary
of the P-disk.)
- If m is a P-line and A is a P-point, why is the inversion B of A in the
support circle of m also a P-point? (In other words, why is B inside the P-disk
and not outside?
- Explain what happens to the shared angle figure in non-Euclidean geometry:
given lines OA and OB, and points A' on OA and B' on OB with OA'/OA = OB'/PB,
is the triangle OA'B' similar to OAB? Explain your answer.
General Hyperbolic
- Define the defect of a triangle. How big can the defect be? How small?
- State and prove the additive property of defect.
- If a geometry contains a defective triangle, what does then there exist a defective right triangle?
- If a geometry contains a rectangle, why does it have an arbitrarily large rectangle?
- If a geometry contains both a defective triangle and a rectangle, explain how the addition of defect in this figure produces a contradiction.
