Part 1. Orthogonal Circles (30 or 40 minutes)
The goal of the first part of the lab is to consolidate the ideas for constructing orthogonal circles.
Break up into 3 groups of 2 or 3. Each group should do the construction and save it to the server in the Math 487 folder. One member of each group will act as scribe and send an email for (to math487@ms.washington.edu) with the names of the group members and the answers to the questions (the group decides on the answers, not just the scribe. Be sure to show your construction to the other groups.
Group 1.
Given two circles c and d and a point P, construct the circle m through P orthogonal to c and d. The construction should work wherever the circles are moved, intersecting or not, and whether or not the point P is inside or outside the circles.
Questions for email:
- Describe the steps of your construction.
- Tracing Experiments. Test 3 cases. Move the circles c and d to an intersecting position, a tangent position, and a disjoint position, where one circle is inside the other but is not concentric. In each case trace m as you drag P and tell what kind of pencil is traced.
- In the case when the traced pencil is the elliptic pencil, this means that there are two points A and B that are on m no matter where P is moved. Explain why B is the inversion of A in c and also B is the inversion of A in d.
Group 2.
Given a circle c with center O and two points P and Q, construct the circle m through P and Q orthogonal to c. The construction should work whether or not the points P and/or Q are inside or outside the circle.
Questions for email
:
- Describe the steps of your construction.
- Tracing Experiments. Trace m as you drag Q. What pencil of circles is traced (for various positions of P and c). What pencil is traced when P is on c?
- Construct the lines OP and OQ. Observe that OP intersects m at P and P’, where P’ is the inversion of P in c. Likewise, OQ intersects m in Q and Q’.
Group 3.
Given three disjoint circles c, d, e, construct the circle m orthogonal to all three circles. The Sketchpad construction should construct m wherever c, d and e move (except for positions where m is not supposed to exist).
Questions for email
:
- Describe the steps of your construction.
- Tracing Experiments. Try several cases and tell whether m exists. (Be sure that this is because of the geometry and not because of a bug in your sketch.) Cases: (a) no two circles intersect (b) two circles intersect (c) each pair of circles intersects but no point is inside all 3 circles (d) there are points inside all three circles.
Part 2. Images under inversion (the rest of the lab time)
Do Exploration 10.1, Investigation 1. Experiment with several shapes. Skip the Explore More.
Construct the image of a segment as a locus and also the image of an arc as a locus, in Investigation 2 of 10.1. Use these figures to answer the questions at the end of 10.1.
Work through all of 10.2. The big question for this section: Why is the image of a line a circle (except for lines through the center of the inverting circle). Read and understand the proof.