Assignment B (Due Friday 1/7)

This is an assignment to learn but not to turn in. It contains the basics of what you should have in your toolkit to understand well the class on Friday. Be prepared to state the "facts" or answer the questions indicated below during class Friday.

Review and Reflection Problems for Friday 1/7

Definitions

Be prepared to write or state clearly, correctly and completely the definitions of:

• Inversion of a point in a circle
• The power of a point with respect to a circle
• The radical axis of two circles.

LOOK OUT for Typical Pitfalls writing definitions

Sermonette: Writing definitions is not just jumping through hoops. Definitions are just as important as theorems. They carefully give names to concepts that are then related to other concepts. But if you don't understand what the original concept is, then the relationships are meaningless. Definitions are also part of one's basic toolkit. If one is asked something about the compostion of two circle inversions, there is nothing one can do without knowing the definition of inversion.

• A definition is a complete, intelligible sentence. Don't write something that would not be clear to a math student who had not attended the course.
• Use the name of the concept to be defined in the definition. Thus the definition of "perpendicular bisector" is not: "The line through the midpoint that is perpendicular." A correct definition is: "The perpendicular bisector of a segment is the line perpendicular to the segment through its midpoint."
• You can use give names: "A tangent line to a circle with center O is a line through some point A of the circle that is perpendicular to OA."
• What is it? If you are defining something, it should be clear whether the something is a point or a circle or a function or a number or .... It will not work just to define a circumstance: "Inversion is when |OP'| = R²/|OP|" is not the definition of anything.
• Don't leave out vital details. For example the inversion of a point P not only satifies an equation involving distances, it is also on the ray OP (O being the center of the circle). If you leave out the ray, you have not specified the point. (In fact we will later work with another point with the same distance formula but on the opposite ray.)
• Don't show off or try to hedge your bets by adding other true facts that are not part of the definition. A classic: A parallelogram is a quadrilateral with opposite sides parallel and equal. (The definition should not have the "and equal" since that is automatic from the opposite sides parallel.)

References:

• B&B, pp. 261-263 for all 3 concepts
• Ogilvy, pp. 25-26 for inversion, radical axis is defined indirectly in chapter 1
• Sved, p. 32 for inversion; all of Chapter 1 for power and radical axis
• GTC, sec. 9.3

Orthogonal circle and line: If c is a circle with center C and m is a line.  What must be true about the line m for m and c to be orthogonal?

Constructing an orthogonal circle given the center: Given a circle c with center C and a point D outside the circle, how can one construct a circle d with center D so that d is orthogonal to C.