Answer all questions. There is additional space to answer questions on the back of the previous pages. Please label answers clearly as (a), (b), (c) etc. Show your work.
Problem 1
Consider a cube with a square face ABCD. Let P be the center of the face opposite ABCD. Form the pyramid with square base ABCD and vertex P.
(a) What is the dihedral angle between the square base and one of the triangular faces? (The answer can be expressed in terms of inverse sin, cos or tan.)
(b) What is the volume of the pyramid?
(c) Let Q be the reflection of P in plane ABCD. Form a double pyramid with vertices ABCDPQ. List the number and kind of symmetries of this polyhedron.
This is a question about the DWEG model. In the figure is the special point O (the point that is removed in the model), along with a Euclidean circle c and two points A and B on the E-circle. [The Euclidean center C of c is included also.] This E-circle c is also a DWEG circle.
(a) Construct the DWEG line m through A that is a DWEG diameter of c. (Hint: What is the angle between a circle and a diameter?)
(b) Construct the DWEG center D of c. Tell how D is related to O and c.
This is a question about harmonic division.
(a) Define what it means for CD to divide AB harmonically.
(b) State (carefully) the relationship between harmonic division and inversion.
(c) In the figure are collinear points A, B, C. Construct a point D so that CD divides AB harmonically.
In this figure is a circle c with center C and a point D outside the circle. Also there is a secant through D intersecting the circle at P and Q.
(a) Construct a circle d with center D so that d is orthogonal to c.
(b) If the distances |DP| = p and |DQ| = q and |CD| = u, what is the radius of circle d in terms of (some of) these quantities?
(c) Construct as efficiently as possible the image of circle d by inversion in circle c. Also construct the inversion D' of D in c.
