Math 445 Quiz 1

Problem 1

Given 4 points A, B, C, D

a)      State what is means to say that C and D divide AB harmonically.  Watch your signs.

b)      True or False?  If C and D divide AB harmonically, then A and B also divide CD harmonically.

Problem 2

Suppose that c is an Apollonian circle of A and B.  Let O be the center of c and let P and Q be the points of intersection of c and line AB.

a)      For the five points A, B, O, P, Q, write down all the relationships where one pair of points divides the other pair harmonically. (The number of lines below does not correspond to the number of answers.)

______________  divides ______________   harmonically

______________  divides ______________   harmonically

______________  divides ______________   harmonically

______________  divides ______________   harmonically

______________  divides ______________   harmonically

b)      In this figure, is B always the inversion of A in c?

c)      True or false?  In this figure, any circle through A and B is orthogonal to c.

d)      True or false?  For any circle d through A and B, P is the inversion of Q in d.

Problem 3

Given a circle c with center O and radius r, let A be a point not on c.  Denote by A' the inversion of A in c.

Explain why any circle through A and A' must be orthogonal to c. Your explanation should include a clear statement of what "inversion" means and what "orthogonal" means, as well as clear statements of key theorems that you use.

Your imagined reader is a student who knows well Euclidean geometry (as in Math 444) but is unfamiliar with inversion.