Some Review ideas for the quiz

Know all the important definitions (Write a definition precisely. Write a complete sentence. Explain all your cast of characters.):

• Apollonian circles
• Harmonic division
• Power of a point
• Radical Axis
• Inversion of a Point

Example of an unsatisfactory definition (half credit for the right equation):  The inversion of P is P’ where |OP’| = r^2/|OP|.

Problems with this "definition":  What is O? What is r? What is P?  Does this work for all P? Can P’ be anywhere with this distance from O? (Maybe OP’ is perpendicular to OP?)

Constructions

• Construct an Apollonian circle of A and B through P.  Or construct the Apollonian circle of A and B with ratio 2/3 (for instance).
• Given A and B and C, construct D so that CD divides AB harmonically.
• Construct the radical axis in all scenarios.
• Construct the inversion of a given point in a given circle. 
• Also construct the inversion of a line or a circle.
• Be able to construct orthogonal circles in all the scenarios.

Theorems

• The definition of an Apollonian circle does not say it is a circle.  How did we prove it?  What theorem about angle bisectors did we use?  How did we prove that?
• The definition of a power of a point does not involve tangents.  How do we get from the definition to tangent lengths?
• The definition of the radical axis does not say it is a line.  How do we prove that?
• Why is the radical axis of two intersecting circles the extended common chord?
• If a circle c is orthogonal to two circles d and e that intersect at A and B, why is c also orthogonal to any other circle through A and B.