Inversion Review

Inversion Review
Definitions

Orthogonal Circles

Inversion

Power of a Point

Radical Axis

Radical Center

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 with Inversion

The basic constructions on orthogonal and tangent circles have been listed well in earlier review material.

In addition, there can be questions that require the interpretation of problems in the DWEG model or the Poincare model as constructions using Euclidean circles.

The key here is to first determine clearly what the construction is, before leaping into the construction.

Examples (In DWEG or P-model, all points, concepts, etc. are taken as beging in the model, not Euclidean plane)

In DWEG, given two points A and B, construct a rectangle ABCD.

In P-model, given two points A and B, construct the perpendicular bisector of AB. Also, construct the circle with center A through B.