Assignment 5. Math 444 Autumn 2000

Assignment for Week 5 (Due Monday, 10/23)

Reminder: Major Quiz #1 is Friday, October 20

Read B&B Chapter 5

This chapter contains some basic theorems about circles, especially chords and inscribed angles, as well as some properties of inscribed polygons, circumscribed polygons, and regular polygons. Also look in the Construction Chapter in B&B as needed.

Assignment 5 (50 points)

5.1 Midpoint Quadrilaterals (10 points)

For a quadrilateral ABCD, prove that the midpoint quadrilateral of ABCD is a parallelogram.

If ABCD is a kite, what kind of polygon is the midpoint quadrilateral? Prove your assertion.

(Note: In a problem of this kind you should give the best or strongest possible answer whether or not it is spelled out each time. For example, it is true that the polygon is a parallelogram, but this is not a satisfactory answer.)

5.2 Inscribed and Escribed circles of a triangle (10 points)

Draw a triangle ABC (not a special triangle) and construct the incircle and the 3 excircles.

If the centers of the excircles are A', B' and C', also construct the altitudes of triangle A'B'C; and explain the relationship of these altitudes to the original figure. (Note: You can do this with Sketchpad in lab on Wednesday. Print it out.)

5.3 Two-edged ruler figure (10 points)

Suppose you are given two parallel lines m1 and m2 which are distance d apart. Let n1 and n2 be another pair of parallel lines that are the same distance d apart, but which are not parallel to the first pair. Then the lines will intersect at four points, forming a parallelogram (by definition!). Prove that this parallelogram is a rhombus.

Note: You can such a figure easily with a ruler alone. First draw a pair of parallel lines with a ruler by placing the ruler on the page and drawing the lines formed by each edge.

5.4 Circumscribing circles given 3 equal segments and 2 equal angles (10 points)

In this problem and the next one, we will consider a figure made of 4 points A, B, C, D and 3 segments in which angle ABC = angle BCD and these 3 segments have equal length: |AB| = |BC| = |CD|. Two examples of such a figure are figure below.

Prove that there is a circumscribed circle that passes through the four points ABCD. (Hint: How do you construct its center?)

Construct this circle in two such examples.

5.5 Special Midpoint Quadrilaterals (Repeat of 4.6) (10 points)

(a) Consider this statement: If the midpoint quadrilateral of quadrilateral ABCD is a rectangle, then ABCD is a kite. Prove it if it is true. If it is false, give a counterexample and, if you can, explain for which quadrilaterals the midpoint quadrilateral is a rectangle.

(b) Consider this statement: If the midpoint quadrilateral of ABCD is a rhombus, then ABCD is a rectangle. Prove it if it is true. If it is false, give a counterexample and, if you can explain for which quadrilaterals the midpoint quadrilateral is a rhombus.