Lab 1: Inversion and Orthogonal Circles

Much of this lab will use sections of Chapter 9 of GTC. Also, notice that the term "pencil of circles" is used and partly explained in GTC. Pencils are special sets of circles that will be explained more fully in class.

Section 1. Inversion of a Point and Orthogonal Circles

Part A. Construction of circle orthogonal to circle c at a given point and through a point A not on c.

• In a new file, carry out the construction on page 152 (in Exp. 9.1 Investigation 3). Give the name c to the circle with center B through C.
• Make a tool that will take points B, C and A and construct the circle with center F. Notice that D is not a given for the tool.
• Duplicate Page 1 to make Page 2 in the same file. Apply the tool to points A, B, C in the proper order to construct a second circle through A orthogonal to the circle c.
• Notice that this tool creates a new point D each time. Use the tool again to make a third and a fourth circle through A.
• Observe that all the points pass through A and a second point that turns out to be A', the inversion of A in c. Also all the centers of these circles lie on a line (the perpendicular bisector of A and A').

Part B. Tracing orthogonal circles

• Duplicate Page 1 again to make Page 3.
• With this construction make the experiments on pp. 155 and the top of 156 in Investigation 9.2. The goal is to visualize and understand what set of orthogonal circles looks like and where the centers are, so that they can be constructed.

Part C. A first construction for inversion of points

• The definitions of inversion and orthogonality that you already know are given on page 158.
• Carry out the construction and experiments on pp. 159-160. Read the definitions at the bottom of page 160.

Part D. Building a first tool for inversion + radical axis

• Construct a figure like the one on page 159 but without the points Q and Q' (you can just take the figure you have and delete or hide the points. Also Hide the points F and D, the line OA, and the circle through A. What should be left is the points O, R, A, A', the circle with center O and the line through F that bisects AA'.
• Now Select All and Make New Tool. Call the tool: Inv Pt + Radical Axis.
• Try it out in a figure with only a circle and a point A. The tool should construct the inversion A' and the radical axis.

Section 2. Orthogonal Circles through Points

Part E. Orthogonal Circle Given Two Points + One Circle

• Carry out the Construction and Experiment of Investigation 1 of 9.4 on pp. 165-6.
• Notice several important points. If A' and B' are the inversions in the circle of A and B, then the new circle can be constructed either as the circumcircle of ABA' or the circumcircle of ABB'. The four points are automatically all on the circle once 3 are. Why?
• Tool "Orthocircle by 1 circle + 2 pts": Hide some construction lines and make a tool that will take O, R and points A and B and construct the circle through A and B that is orthogonal to the given circle. (The tool on the disk is for GSP version 3.)

Part F. Orthogonal Circle Given One Point + Two Circles

• Carry out the Construction of Investigation 2 of 9.4 on page 166-7.
• Tool "Orthocircle by 2 circles + 1 pt": Make a tool that will take B, C, D, E and A and construct the circle through A orthogonal to circles BC and DE7. (The tool on the disk is for GSP version 3.)
• Carry out the Experiment of Investigation 2 of 9.4 and especially fill in the table corresponding to Q2.

Section 3: Tracing Orthogonal Circles and the Radical Axis

Part G. Tracing the Circles Orthogonal to Two Given Circles

Next, carry out Construction -- Tracing Circles Orthogonal to Two Circles but using these CHANGED INSTRUCTIONS for VERSION 4.

• In a new sketch, draw circles circles c1 and c2. Also draw a point A as a point on c1. Then apply your tool "Orthocircle by 2 circles + 1 pt" to the two circles and A. You should get a circle d through A that is orthogonal to both c1 and c2.

Experiment 1. Trace the centers to see the Radical Axis

• If P is the center of circle d1, then trace P as you drag or animate A around c1. What is the shape of the trace of P. You can also select P and A and construct Locus to get a dynamic version of this set. Try this and then move the two circles so that they are (1) disjoint and exterior to each other (2) disjoint and one is interior to the other (3) tangent to each other and (4) intersecting in 2 points. Notice where the locus of P is in each case. This locus is called the Radical Axis of c1 and c2.

Experiment 2. Trace the circle d to see a "pencil" of circles

• The goal of the experiment is to visualize all the circles orthogonal to c1 and c2. There are two ways to do this.
  • One is to drag or animate A (just A, not the circles) and trace circle d.
  • The second is to select the circle d and A and Construct Locus to construct the locus of circles.
  • Or you can try both.
• When you have this set up, use your experiments to figure out the answer to the Matching Question Q2 at the end of this section.

Things to Think About

• The d circles trace a family of circles with centers all on some special line (it is called the radical axis of c1 and c2).
• The center of the circle d has the same value of the power function with respect to c1 and also c2 (this function is the square of the radius of d).
• The radical axis is the set of points P for which the two power functions are equal (this will be proved in class or on assignments).

Part H. Tool for the Radical Axis of Two Circles

Continue with the figure above with circles c1 and c2, the point A on c1 and the circle d through A with center P. (Hide or get rid or traces and loci.)

• Construct the radical axis as the line through P that is perpendicular to the line through the centers of c1 and c2.
• Hide the point A, the circle d and the point P. Then select all and make a tool. It should take as givens four points to construct c1 and c2 and the radical axis.

Experiment

• Draw 3 circles, c1, c2, and c3. Construct the radical axis of each of the 3 pairs of circles.
• Check that the radical axes are concurrent or parallel. Construct the point of concurrence O. This point is called the radical center of the 3 circles.
• Construct a point T on c1 so that line OT is a tangent line to c1 (moving the circles so that O is outside the circles if necessary). Construct the circle d through T with center O. This circle d should be orthogonal to all 3 circles. It should in fact be the only such circle! Explain.
• Are there cases where the circles intersect and the circle d exists? Are there cases when O is inside two circles and not inside the others? What happens to d in this case? (Hint. What is the power of each of the circles at O?)