Lab 4: Circles, products and angles

This lab is devoted to a review and some preliminary work on circle geometry that will be key to the next topic in the course, which is geometry of orthogonal circles and inversion.

Part A: Secant lines and products

Download the GSP file lab04.gsp here

Exploration A1: Observing the phenomenon

On page 1 is a circle with radius r and a point P, and also a line through P intersecting the circle at A1 and A2.

Denote the distance OP by d.

a) Compute the distances PA1 and PA2 and also the product of these two distances. Now drag A1 around the circle. How does the product vary?

b) If T is a point on the circle so that PT is tangent to the circle, how is the distance PT related to this product (move A1 so that PA1 approximates PT).

c) If A1 is moved so that PA1 is a diameter, how is the product related to r and the distance OP? Can you write a formula for this?

d) Finally, move P inside the circle. Again observe the product and again find the formula for the product in terms of r and OP.

e) What happens to the product and your formulas when P is on the circle?

Report Out A1 (oral)

How does the product behave? What formula(s) for the product have you found?

Exploration A2: Explaining the phenomenon

In the same sketch, construct a second line through P intersecting the circle in B1 and B2.

To explain the constancy of the product PA1*PA2 from A1, it is sufficient to show that for any B1, PA1*PA2 = PB1*PB2. (There is also the special case of the tangent line that must be shown separately.)

Report Out A2 (written)

Write a proof of this constant product fact, both in the case of P outside the circle and P inside the circle.

• A proof can be found using similar triangles and what you know about angles in circles. You have done some of this before in 444 in problem sets. But in 445 this will be a key tool, so we need to review this and bring it to the forefront.

Exploration A3: Naming the phenomenon

For a circle c with radius r and center O, and any point P in the plane, the function c(P) = OP^2 = r^2 is called the Power Function of P with respect to c (or sometimes the value of the function is called the Power of P with respect to c.

• Explore how this power function is related to your products in the cases above, both for P inside and outside the circle.

Report Out A3 (written)

How is the power function related to distance associated with secants and tangents? When is this function negative? How can you make sense of this with respect to the phenomenon of A1?

Part B: Angles between Circles

This one is a short observation to understand a definition. Go to Page 2 of the Lab 4 GSP file and experiment with making the scale factor very big.

a) How does this explain/justify the definition of the angle(s) between two circles as the angle(s) between the tangent lines at a point of intersection?

b) Why was angle(s) written and not angle?

c) If the circles intersect at two points, does it matter which point of intersection you choose? Why?

Part B Report Out (oral)

Be prepared to discuss or explain the points in B above.

Part C. Orthogonal Circles

Exploration C1: Tangent as a circle radius.

Go to page 3 of the GSP file and construct the tangent and the circle as directed. What is the angle between the circles?

Part C1 Report Out (written)

Write down what the angle is and why.

Exploration C2: More construction of orthogonal circles

Work through the pages copied from Geometry Through the Circle.

Part C2: Report Out (oral and construction)

Answer the questions on the pages and learn the constructions. Be prepared to answer or describe orally or to carry out the constructions with straightedge and compass.