Inversion Study Sheet

Inversion Study Sheet

Definitions

Discussion Problems and proofs

Define inversion of a point P in the circle c and prove that any circle through a point and its inversion is orthogonal to c.

Prove that the inversion of a line in a circle c is a circle or a line [including infinity in the lines to get the complete figure]. Note: A proof is in Ogilvy and also in Sved. When the line is not through the center of c, there is ONE proof that works for all cases. When you construct the image, there may be different strategies that are more efficient in different cases, for example, when the line intersects the circle c. Be sure to distinguish between what you do for construction, when you KNOW the image is a line or circle, and the proof, when you do not yet know it.]

Let c be a circle with center O and let P and Q be points with inversions in the circle P' and Q'. Prove that triangle OPQ is similar to OQ'P'. However, discuss and sketch the images of the SIDES of triangle OPQ and tell whether they are the same as the sides of triangle OQ'P'.

Define the radical axis of two circles and prove that, given 3 circles, the 3 radical axes of pairs of circles are concurrent or parallel [with a special case]

Explain how two disjoint circles can be inverted to concentric circles and why your method works. [This could also appear as a construction problem.]

Given two intersecting circles c1 and c2 and a circle d orthogonal to both, explain why a circle c3 in the same pencil as c1 and c2 is also orthogonal to d. [Two possible kinds of pencils.]

Constructions

Know constructions for inversion image of a point, a line, a circle.
For a point P, know how to construct (when possible) a circle d with center P so that d is orthogonal to a given circle c1. Or to two given circles c1 and c2. Or to three given circles c1, c2, c3.
For 3 objects, each of which is a point or a circle, construct a circle d that passes through the objects that are points and is orthogonal to the objects that are circles (compare with Lab 5).
Construct radical axis of any two circles.
Given two circles and a point P, construct the circle through P that is in the same coaxal family as the two circles.
Given two disjoint circles, construct a circle c so that the inversion image of the two original circles are two concentric circles.
In DWEG geometry, construct lines that are parallel, orthogonal, perpendicular (all in D sense). Know how to construct a rectangle. Also construct the D circle with D-center A through D-point B. (What if one of these points is infinity?