Math 445 Lab 2

This lab is based some sections of Chapter 9 of Geometry through the Circle (GTC). We have supplemented these activities and altered the order of activities in GTC to give a better fit with the presentation in Sved, Chapter 1.

A. Introduction to Orthogonal Circles

For the first 2 experiments, use the already-made Sketches on the lab server rather than constructing them. These two experiments should be done briskly.

  1. Using the file 9.1.1 Angle Between Circles, perform the experiment at the top of page 148 of GTC. Drag the point G to observe the dilated arcs. This dilation shows how from a "close up" view arcs look like rays; this motivates the definition of angle between circles.
  2. Using the file 9.1.2 Rotate Circle w Arc Arws, carry out the experiment on page 149 of GTC and answer questions Q1-Q4 to get a sense of orthogonal circles, especially the relationships of tangents and radii.
  3. Construction of a circle d orthogonal to circle c, given the center D of d. Observe the figure on GTC p. 150, but instead of following the directions, construct the figure with different givens. Start with circle AB and point D and construct point C and tangent segment DC using the tangent construction from last quarter. Measure the length of DC. Hide construction lines and make a script Ext Tan Segment that construct DC and the measurement. Construct the circle DC and note that the two circles are orthogonal.

    B. Pencils and Radical Axes

Note on terminology: What is called a two-point pencil in GTC is called an elliptic pencil in Sved. The double-point pencil in GTC is a parabolic pencil. The third kind of pencil that is introduced later as a 0-point pencil is a hyperbolic pencil in Sved.

  1. Elliptic Pencil. Construct an elliptic pencil as illustrated on page 153 of GTC, Exploration 9.2, Investigation 1.
  2. Radical Axis for an Elliptic Pencil. Continue your construction from 4 to get the figure on page 16 of Sved. You can construct a single circle and tangent segments and trace them get the figure: Construct the center C of the circle as a point on the perpendicular bisector of segment AB. Construct P as a point on line AB. Then, construct the tangents PT1 and PT2 from P to circle CA using the script Ext Tan Segment from 3. Now drag point C and observe that the lengths of the tangents stay the same. If you trace the points T1 and T2 as C is dragged, what is the trace? Why?
  3. Parabolic Pencil. Repeat the construction of 4 and 5 for a parabolic pencil. (You can make a separate construction or you can approximate it by dragging B to be very close to A.)

    4. C. Radical Axis of a Circle and a Point

  4. Key construction. Carry out the construction on page 152 of a circle through points A and D that is orthogonal to circle c (where A is not on C and D is on c). Make the script. (This is the same as problem #8 of Chapter 1 of Sved.)
  5. Orthogonal Circle Trace. Using the construction from 7, carry out the Investigation 2 of 9.2 on pages 155-6 of GTC.
  6. Radical Axis Script for Pt and Circle. In the construction on page 155, construct the line through F perpendicular to line AB. From your experiment, you have observed that this line does not depend on the choice of D but only depends on A and the circle BC. This line is called the radical axis of the point A and the circle BC. Make a script that will construct the radical axis, given A, B, and C (hide the construction items such as D, F and the circle DF). This definition of radical axis is an extension of the definition of the radical axis of two circles in Sved, where we consider A to be a circle with radius 0.

    7. D. Constructing Orthogonal Circles through Points

  7. Circle through two points orthogonal to a circle. Carry out Investigation 1 of 9.4 of GTC. This is Sved, Chapter 1, #10.
  8. Circle through one point orthogonal to two circles. Carry out Investigation 2 of 9.4 of GTC. (Note: the title of this investigation is missing in the first printing of GTC, but you should have the second printing unless you got an old copy. In either case, this investigation runs from the second page of 9.4.) Continue until you reach the definition of radical axis of two circles. Compare this definition with that of Sved. This is Sved, Chapter 1, #12.

    9. E. Constructing the circle orthogonal to 3 circles

  9. Sved, Chapter 1, #11. Hint. The center of the circle must be on all 3 radical axes.