Final Exam

NAME________________________________

Problem 1: Similarity and Inversion

Let c be a circle with radius R and center O. Suppose that the inversions of two general points A and B in c are A' and B'.  Two (or more) of the triangles that can be formed from the points A, B, A', B' O are similar. 

Write a pair of similar triangles.

Prove the similarity.

State briefly the key idea of how these similar triangles are used to prove that the inversion of a line not through O is a circle (with one point exception)

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Problem 2. Harmonic point calculation

Suppose A, B, and C to be points on the real number line. If A = 1, B = 10, C = 46.  For what point D does CD divide AB harmonically?

D = ____________

Problem 3: Radical Axis

Define the radical axis of two circles c and d.

The radical axis of two circles is a line.  Is this part of the definition or not?  Yes ____ No _____

Problem 4. Ceva calculation

Given triangle ABC and points A' on BC, B' on CA and C' on AB.  Suppose the ratio BA'/A'C = 4/3 and ratio CB'/B'A = 5/2.

(a)    What are the barycentric coordinates (with respect to ABC) of P, the intersection of AA' and BB'?

Barycentric coordinates of P = __________________________

(b)   If AA', BB' and CC' are concurrent, what is AC'/C'B?

AC'/C'B = ____________________________

(c)    If A = (1, 1, 0), B = (0, 2, 1), C = (1, 1, 1) what point is P?

P = ______________________________________________

Problem 5. Circle on a sphere

Let S be the sphere with center (0,0,0) and radius 1. The plane p with equation x+y+z = 1 intersects S in a circle c. 

The circle c is a circle in the plane p, but it can also be considered as a spherical circle of points whose spherical distance from a center on the sphere is constant.

(a)    What is the center C of this circle c in the plane p?

C = __________________

(b)   What is the radius of c in plane p?

Radius = ________________________

(c)    What is a center D of this circle on the sphere S?

D = __________________________

(d)   What is a spherical radius of this circle? (angle measure or given as an inverse trig function)

Spherical radius = ___________________________________

Problem 6. Lines and projections

(a)    Given point K = (0,0,1) and point P = (a, b, c), write the parametric formula for line KP.  Choose the parameter t so that t = 0 at K and t = 1 at P.

Parameterization: ___________________________________

(b)   Use the parameterization above to write a formula in terms of (a, b, c) for the intersection Q of the line KP and the plane with equation z = 0.

Q = _______________________________

Problem 7: Stereographic Projection

·        If A is 45 degrees from N in spherical distance, calculate the distance from O to B.

Distance (in terms of R)  _____________

Problem 8: Construct a common perpendicular in the P-disk

This figure shows two P-lines in the Poincaré disk model (one of the P-lines is a Euclidean segment).  Construct the P-line m orthogonal to both of these lines.

Problem 9: Constructions of Orthogonal Circles

Write briefly and clearly your main steps. O1 and O2 are the centers of c1 and c2.

(a)    Construct and label the circle m through A and orthogonal to c1 and c2. 

(b)   Construct and label the circle d through A that is in the same coaxal family (pencil) as c1 and c2. 

(c)    Construct and label a point N so that for a circle n with center N, the inversion images of c1 and c2 are concentric circles. (Just construct N.  Do not invert!)

Problem 10: Construction of a Central Projection

Do not write in this table.  Reserved for Grader.