Study Problems for Midterm 2
First, begin with the study problems for Quiz 2.
Harmonic division and Apollonian circles
- Definition of harmonic division - ABCD.
- Given collinear A, B, C, how can one construct with a straightedge only
a pont D so that CD divides AB harmonically? How can the correctness of this
be proved from Ceva and Menelaus?
- 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
Review the study problems for Quiz 2.
- 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 DWEG
Lab 7.
- 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 the
P-model Lab 8, including the ones below.
- Given a P-line AB and a P-point C, construct the 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.
- 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 Thales 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.
Projective Geometry
- What is central projection and why does a line project
to a line?
- State the theorem of Pappus. State the theorem of Desargues.
- Tell how the theorem of Desargues proves that if triangles
ABC and DEF have corresponding sides parallel, then the lines AD, BE, and
CF are either concurrent or parallel.
- Apply Desargues theorem to this figure to find 3 collinear
points.
- Explain why these two figures are the same from the point
of view of projective geometry (i.e, how one can be projected to the other?).
In the second figure the lines that appear parallel are parallel. You should
label the second figure with the same letters are the corresponding points
in the first figur
- Construct the polar of point P with respect to a circle c.
