Constructions of P'

There are several ways to construct the inverse of a given point P with respect to gamma. The Construction Menu on the previous page contains links to six of these constructions created with Geometer's Sketchpad. These files contain step-by-step details in the construction and a brief explanation of why the constructed point P' is the inverse of P with respect to gamma. These sketches are intended to be used as guides for the teacher. Questions and activities for students are listed below.

1. Construct P' for each of the following cases. After each construction, prove why the method works.

Method 1a: Given circle gamma with center O and radius r, point P not on gamma (and not equal to O). Construct M on the intersection of ray OP and gamma. Construct N such that PN = r and N is on line OP. Choose a point K (not equal to M) on gamma. Construct ray OK. Construct the line through N that is parallel to KP. This line will intersect ray OK at a point, call it point L. Now KL=OP'. Now construct point P' on ray OP. See the figure below.

Method 2: Given circle gamma with center O and radius R, point P outside of gamma. Construct a circle with center P where O is on the circle. This circle intersects gamma at point E. Now construct a circle with center E where O is on the circle. The intersection of this circle and ray OP is P'. See the figure below.

Method 3: Given circle gamma with center O and radius r, point P outside gamma.Construct M, the midpoint of segment OP. Construct a circle with Center M that passes through O and P. This circle intersects gamma at two points. Call them E and F. Construct segment EF. The intersection of segment EF and segment OP is P'. See the figure below.

Method 4a: Given circle gamma with center O and radius r, point P inside gamma. Construct ray OP. Construct the line through P that is perpendicular to ray OP. This line intersects gamma at points E and F. Construct the line through point E that is perpendicular to OE. This line intersects ray OP at P'. See the figure below.

2. For each of the methods above, will the construction work for any location of P. What are the limitations (if any) of each method?

For Method 1a, OP < 2r. For method 2, OP > r/2. For method 3, P must be outside gamma. For method 4, P must be inside gamma.

3. Create a Geometer's Sketchpad custom tool (or tools) that will enable you to construct the inverse of point P for any location with respect to gamma.

In order to make a custom tool, complete the construction and then select the given items (center of gamma, point on gamma, P) and the item to be constructed (P'). Note: this will not work with constructions that require the arbitrary selection or creation of geometric structures. Construction methods 3 and 4a work well to this end.

4. Are other methods to construct P'?

You may wish to introduce Constructions 1 and 4, or an original construction of your own.