This chapter contains some basic theorems abut circles, especially chords and inscribed angles, as well as some properties of inscribed polygons, circumscribed polygons, and regular polygons.

Some of this Chapter has been assigned before, but now you should read the rest. The highlights that you should be sure to know are

• congruence for triangles and why SSA (or ASS) does not work;
• constructions -- triangles from SSS, copying angles, perpendicular bisectors;
• rectangles, parallels and alternate interior angles;
• loci -- points equidistant from two points; points equidistant from two lines;
• the method for measuring the size of the earth according to Eratosthenes and Posidonius;
• Euclid's Postulates and the collapsible compass.

E7.1 Inscribed circle

• BB, p 141. #8

E7.2 Circumscribed Quadrilateral

• BB, p. 142, #9, #11

E8.1 Locus of points equidistant from two lines

• Given two lines m and n intersecting at A, these lines define four angles with vertex at A. Prove that a point P distinct from A is equidistant from m and n if and only if the ray AP is the bisector of one of these four angles.

E8.2 Inscribed and Exscribed circles of a triangle

• 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.)

Let m be a line and let d be a positive number. Let N be the set of points P for which the distance from P to the line m is d. Prove that N consists of two lines, n1 and n2, each of which is parallel to m.

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. Then move the ruler and do this again; you get a figure like the one below.

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 which passes through the four points ABCD. (Hint: How do you construct its center?)

Note: In problem 4.4 you are asked to construct this circle in two examples. You may prefer to do this as you work on 4.3.

Use the same figures as in 4.3.

• Prove that there is a circle which is tangent to the segments AB, BC, and CD. Where are the points of tangency?
• Tell how the center of this inscribed circle is related to the center of the circumscribed circle of 4.3 (and prove it).

• Construct two figures approximately like the examples above (i.e., one has intersecting segments and one does not). Use Sketchpad or a straightedge and compass to construct the circumscribed circle from 4.3 and the inscribed circle from 4.4 for these two figures.

(Note: This is not the construction on page 191 of B&B because the givens are different. Construct with straightedge and compass or Sketchpad; write briefly the steps of your construction and leave the construction marks so that one can follow what you did.)