Textbook: Ogilvy, Excursions in Geometry

Math 445 Assignment 1-09.  Due Wed 1/08

Problem 1. Harmonic Division

This exercise is just practice to check your understanding of the definition of harmonic division and also to give some examples to give a picture of what the concept looks like.

A. Given points A and B and a constant k > 0, use algebra to prove that there is exactly one point C in segment AB so that |AC/BC| = k.

Hint:  Using a ruler and choosing units so that |AB = 1 unit, you can assume that line AB is the number line with A = 0 and B = 1.  Then assume C is number c and solve for c.

B. Given points A and B and a constant k > 0, use algebra to prove that there is exactly one point D in on line AB but not in segment AB so that |AD/BD| = k. [This is true except for one special case of k.  What is this k?]

C. Assuming a ruler with point A = 0 and point B = 1, fill in the table of values of k with the number values of the points C and D according to the ruler.

k

C

D

99

    

4

    

2

    

1

    

1/2

    

1/4

    

1/99

    
   

4
 

1/4

  

Problem 2. 

Review Exercise.  (This will be needed for Wed. class. The explanations and proofs use theorems from 444.)

In the figure below, the circle is an Apollonian circle of E and F with unspecified ratio k. O is the center of the circle and line DE is tangent to the circle at T.

Given these facts, let a = angle EOT.  Then answer the following questions.

·        Since the circle is an Apollonian circle, what is special about rays TG and TH in triangle ETF?

·        If angle EOT = a, what is angle GTE?  Why?

·        If angle EOT = a, what is angle STG? Why?

·        If angle EOT = a, what is angle HTS? Why?

·        If angle EOT = a, what is angle DTH? Why?

·        Use some of your answers above to prove that ST is perpendicular to line EF.

·        Now that we know that angle EFT is a right angle, use similar triangles to find a relationship among OE, OF, OT.  (Note: OT = r = radius of the circle.)

Problem 3.

Given the same figure, but this time we are given that line ED is tangent to the circle at T and also given chord ST is perpendicular to line EF.  Then prove all the angle relationships from Problem 2 and use this (along with the angle bisector theorem) to prove that the circle is an Apollonian circle of E and F.