Math 445 Assignment 6

Math 445 Assignment 6 (due Monday, 2/12)

Note: Many of the constructions will be done in LAB!

Problem 6.1: Exposition of some basic orthogonal relations (20 points)

This is a writing problem somewhat different either from a proof or from a short-answer question. What is wanted is a clear set of statements and explanations that set up a train of important ideas in the geometry that we are studying now. You can think of it as an executive summary for a friend whom you are prepping for a test. It should be clear and not vague. You do not have to put in proofs to support every statement you make, but some reasoning that connects ideas is explicitly called for. Draw figures and label them to make the explanations briefer and clearer.

Explanation 1: Given a circle c and a point P outside the circle, state the definition of Power of the point P with respect to the circle, and explain how this quantity can be computed by any of the following:

a tangent to the circle through P
the diameter of the circle through P
any secant through P intersecting the circle in two points.

Explanation 2: To the figure above, add the circle d with center P that is orthogonal to c.

Tell how the radius of the circle d is related to the power of P with respect to c.
If a secant through P intersects the circle c in points A1 and A2, explain why A2 is the inversion of A1 in d.
Explain why a consequence of this relationship is that the inversion of the circle c in d is c itself.

Explanation 3: Given a new figure with a point Q and a circle e centered at Q. Let B be a point not on e and let B' be the inversion of B.

For any circle f through B and B', what is the power of Q with respect to f. (Specific answer is needed; it should explain why the power is the same for any such f.)
For any circle f through B and B', what is the length of a tangent segment from Q to f. (Specific answer is needed; it should explain why the power is the same for any such f.)
Explain why the circle e is orthogonal to any circle f through B and B'.

Problem 6.2: Radical axis is a line (10 points)

Let P be a point (x,y) in the coordinate plane. If c1 is the circle with center (a,b) and radius r and c2 is the circle with center (c,d) and radius s, write down an expression in x and y (and the other given quantities) for (1) the power of P with respect to c1 and (2) the power of P with respect to c2.

State the definition of the radical axis m of c1 and c2 as the locus where the two powers are equal. Write down and simplify an equation in x and y for the radical axis.

Show that this equation is the equation of a line perpendicular to the line of centers of the circles, so the radical axis is proved to be a line perpendicular to the line of centers. (Are there any special cases that are exceptions to this statement?)

Explain why the fact that the radical axis is a line implies that the radical axis of two intersecting circles is the common chord extended (or the common tangent line if the circles are tangent).

Problem 6.3: Radical center and a shortcut (10 points)

Part (a) Proof of concurrence of radical axes.

Given 3 circles, each pair of circles has a radical axis. Prove that these 3 radical axes are concurrent. The point of concurrence is called the radical center.

Comment 1. There is a special case when this is not precisely true. Make note of it.

Comment 2. The logic of the proof is the same as the logic in the proof of concurrence of perpendicular bisectors of sides of a triangle, so you may wish to refer back to that for inspiration.

Part (b). A shortcut for constructing the radical axis of two non-intersecting circles.

Given two non-intersecting circles c and d, draw any circle e which intersects c in two points and also intersects d in two points (there are an infinite number of choices for e). Let lines m and n be the common chords (extended) of e and c and of e and d.
Explain why the intersection E of m and n is a point on the radical axis of c and d.
To construct the radical axis p of c and d, then construct the perpendicular through E to the line of centers of c and d. (Alternatively, one may construct a second point F on the radical axis in the same way that E was constructed; then p = line EF.)

Problem 6.4: Radical axis and orthogonal circles (construction, 10 points)

This problem calls for 3 examples of the same construction. You may wish to construct these figures with Sketchpad and print them out.

Description of construction: Draw a figure with two circles c and d. Construct the radical axis of the two circles. Choose a point P on the radical axis and construct a circle p with center P so that p is orthogonal to c. Observe that p is automatically orthogonal to d.

Construct such a figure when the circles c and d intersect at two points. Question: Do there exist any points P on the radical axis where no circle p exists? What can you say about the power of P at such a point?
Construct such a figure when the circles c and d do not intersect, and each circle is exterior to the other, but of different radii (i.e., the circles are "side by side"). Notice that the radical axis is NOT the perpendicular bisector of the segment connecting the centers of c and d.
Construct such a figure when the circles c and d do not intersect, and circle is interior to circle d, but visibly not concentric (i.e., circle c is "off-center").

Problem 6.5: Construction of Orthogonal circles using tangent line (10 points)

Construct each of the following in an example.

Given a circle c and points P and Q on c, construct the circle p through P and Q that is orthogonal to c.
Given a circle c and point Q on c and P not on c, construct the circle p through P and Q that is orthogonal to c.

Problem 6.6: Construction of Orthogonal circles using inversion (10 points)

Construct each of the following in an example.

Given circles c and d and a point P not on the circles, construct the inversion P' of P in c and the inversion P'' of P in d. Then construct the circle p through P, P', and P''. Explain why p is orthogonal to both c and d.
Given a circle c with center O and a points P and Q not on the circle, construct the inversion P' of P in c. Then construct the circle q through P, P', and Q. Explain why q is orthogonal to c. Also explain why the inversion Q' of Q in c is the other intersection of line OQ and circle d.