Lab 6: Inversion and Orthogonal Circles

Lab 6: Inversion and Orthogonal Circles

This lab will consist of sections of Chapter 9 of GTC. But first, make a quick secant figure as a review.

Part A. Secants and Power

Draw a circle with center A through B and also draw a line CD. Intersect the line with the circle to get points E and F. If you leave C fixed and drag D, then line CD is a secant through C intersecting the circle in points E and F.

Measure |AB| and |AC| and use the calculator on these measurements to compute the power of C with respect to the circle. Move C around and observe the locations where C is positive, zero, and negative.

Now measure |CE| and |CF| and calculate the product of these two numbers. Compare with the power of C. Drag D, leaving C fixed, and observe that this product does not change.

Construct the tangent from C to the circle. Let T be a point of tangency on the circle. Measure CT and compute the square of this number. Check that this also is the power when C is outside the circle.

Drag C inside the circle, notice that that tangent length disappears and the product |CE||CF| now has the wrong sign for the power. [This could be fixed by multiplying by sgn(CE/CF) if you wanted.]

Part B. Construction of circle orthogonal to circle c at a given point.

Carry out both constructions in Investigation 3 of 9.1 of GTC (pp. 150-1).

Carry out the construction on page 152 (same Investigation) and make the script.

Part C. Definition of Pencils

Read the definition of two kinds of pencils on page 153. Also read the note below. Answer the questions but do not actually carry out the constructions.

Notation Note RE page 153 and later: GTC tries to avoid technical terminology in the definition of pencils, but the other texts use the correct terminology. A two-point pencil in GTC has standard terminology = elliptic pencil. A double-point pencil = parabolic pencil. A 0-point pencil = hyperbolic pencil

Part D. Tracing Circle Orthogonal to a given circle.

Do the construction and experiments on pp. 155 and the top of 156 in Investigation 9.2.

Part E. Some constructions for inversion of points

After reading the familiar definitions and statements on page 158, carry out the construction and experiments on pp. 159-160. Read the definitions at the bottom of page 160.

Carry out the construction of Investigation 2 on pp. 161-2. Make the script.

Carry out the construction of Investigation 3 on pp. 162-3. Make the script.

Carry out the first construction of Investigation 4 on pp. 163-4. Make the script.

Part F. Circles orthogonal to circles

Carry out the construction of Investigation 1 of 9.4 on pp. 165-6.

Carry out the construction of Investigation 2 of 9.4 on pp. 166-7.