1. Orthogonal Circles and Inversion

Two circles are called orthogonal if their tangents (or radii) at a point of intersection are perpendicular to each other.

The investigation below relates orthogonal circles to an algebraic property of geometric inversion.

a. Use the free-hand circle tool to draw a circle C. Label the center S and the control point B.

b. Draw a point P in the interior of the circle. (P can be random in the whole plane.)

c. Construct any point Q on circle C.

We'd now like to construct a circle C' that is orthogonal to C and that passes through point P. How to do it? Let's make it so that it intersects C at point Q. Now, one thing is that if it is to pass through both P and Q its center must be on the perpendicular bisector of segment PQ, so

d. Construct the segment PQ, the midpoint M of PQ, and the perpendicular L to PQ through M.

And, if the two circles are to be orthogonal, their radii, at a point of intersection, must be perpendicular to each other, so

e. Construct the segment, r, joining S and Q, and the perpendicular, N, to segment r through Q. The symbol r will also represent the radius of the circle C.

f. Construct the point of intersection, S', of line N and line L.

g. Construct the circle, C', with center S' that passes through Q.

Problem 1: Why must the circle C' be orthogonal to circle C?

___________________________________________________________________

___________________________________________________________________

Problem 2: Why must C' also pass through point P?

___________________________________________________________________

___________________________________________________________________

h. Construct the ray R from center S through point P.

i. Construct the point of intersection P' of ray R and circle C'.

Problem 3: Use the definition of power of a point with respect to a circle to express a relationship among SP, SP', and r. (Write an equation.)

Problem 4: Compare the result at the end of Lab 10 to express the relationship between P and P'. (No equation)

Cf. problem #9, page 24 of Journey.

2. Some Reasoning in Journey

The point E in the diagram below has been chosen (somehow!) so that its power is the same with respect to each of the two circles. Symbolically:

(Click on the diagram to go to an interactive version.)

The point P is any point on the perpendicular to the line joining the centers, O1O2, through E. The objective is to show that the power of P is the same with respect to each of the two circles, and hence line PE is the power line, I mean radical axis, of the two circles.

Step 1: Use the definition of power to express the power of P with respect to C1, P(C1), in terms of the distance PO1 and the radius r1.

P(C1) =____________________

Step 2: Use the Pythagorean Theorem on triangle PO1E to express the right-hand side of the equation in Step 1 in terms of EO1and h. (Repeat the Step 1 equation below and continue with a second equation for this step.)

P(C1) =____________________=______________________

Step 3: Now use the definition of power to express the far right-hand side in Step 2 in terms of h and the power of E with respect to C1.

P(C1) =____________________=______________________=_________________

As a result of these three steps, you should have a relationship (equation) between the powers of P and of E with respect to the circle C1. Do three analogous steps using the triangle PO2E to express a relationship (equation) between the powers of P and of E with respect to the circle C2.

Step 1': Use the definition of power to express the power of P with respect to C2, P(O2), in terms of the distance PO2 and the radius r2.

P(C2) =____________________

Step 2': Use the Pythagorean Theorem on triangle PO2E to express the right-hand side of the equation in Step 1' in terms of to EO2and h. (Repeat the Step 1' equation below and continue with a second equation for this step.)

P(C2) =____________________=______________________

Step 3': Now u se the definition of power to express the far right-hand side in Step 2' in terms of h and the power of E with respect to C2.

P(C2) =____________________=______________________=_________________

Finally, use the results of Step3 and Step 3' in light of equation (*) near the top to compare P(C1)and P(C2) .