Math 487 Lab 1

Given two points and a number k >0, by definition the Apollonian locus that turns out to be an Apollonian circle is the set of points P with |AP|/|BP| = k. How can be model this with Sketchpad?
Here is a hint. Given a segment PQ, let R be a point on the segment. Hide the segment so that P, Q and R are showing, and construct segments PR and RQ. Measure the ratio PR/QR. If you leave P fixed and drag Q to change length PQ, then observe the ratio stays the same.
Now, construct a circle with center A and radius PR and another circle with center B and radius QR. Then the intersection points S and T of the two circles will trace out the locus.

2. Constructing the circle using angle bisectors.

In a new sketch, given points A and B and a point P, construct the angle bisectors of angle P in triangle ABP and intersect these bisectors with line AB to give C and D. Then the circle with diameter CD passes through P and is an Apollonian circle.
Make your circle red. Once you have constructed the circle, try tracing the circle as you drag P to get a figure something like Ogilvy, page 18.
Now construct the circle through A, B and P and color it blue. Also trace this circle. Drag P to get a figure something like Ogilvy, page 21.
Check by measuring the appropriate angle that the red circle and the blue circle are always orthogonal.

In GTC, carry out 9.1, Investigations 2 and 3, pp. 149-151 (excluding the Explore More).
Once you have made the circle, trace the circle as you drag point D. Observe that all the circles not only pass through A but also through another point on the other side of the circle from A (this will turn out to be the inversion of A in the circle).
Carry out the last construction of 9.1, on page 152, and turn it into a script as directed.
Construct with Sketchpad a figure like Fig 16, on p. 22 of Ogilvy (you can make one circle through AB and move it, tracing the tangent points)

Carry out GTC 9.3, Investigation 1, pp 158-160.