Assignment 7 ANSWERS

Study Problems – You will be asked to write for 10 minutes on one or more of these questions on Wed.

These are questions 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.

Ref. Sved, Chapter 1.

Explanation 2: State the definition of inversion. 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.

Ref. Sved, Chapter 2. Also, Ogilvy.

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.)

QB QB'

• 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.)

Sqrt QB QB'

• Explain why the circle e is orthogonal to any circle f through B and B'.

By def of inversion, the radius of the circle is also Sqrt QB QB'. Therefore the radius = tangent.

Explanation 4: Explain why the radical axis of two circles is a line.

Sved, Chapter 1, or handout in class using formula for power of a circle in (x,y) coordinates.

Assignment 7 (Due Wed 2/19) Turn this part in.

Problem 7.1: Radical center and a shortcut and a construction

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 of the 3 circles

Answer: This is Sved, Chapter 1, Problem 4. Answer in back of book.

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

If centers of circles are collinear, the radical axes are parallel. If two circles are concentric, there are no points in radical axis.

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) 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.)

Answer: This is Sved, Chapter 1, Problem 5. Answer in back of book.

Part (c) Construction

• Given 3 circles, construct a circle orthogonal to all 3 circles.  Do this for two cases: (i) when all 3 circles are non-intersecting and exterior to each other.  (ii) when two or more of the circles intersect.  Hint:  What must be the center of the orthogonal circle?  Note: There are some arrangements of 3 circles for which there is no circle orthogonal to each.

The center O of the circle is the intersection of any two of the radical axes. If O is inside one of the circles, then it is inside all 3 and there is no such orthogonal circle. Assume O is outside the circles, so the power of the point O is positive. Then by definition of the power, if one constructs tangents from O to each of the circles, all the tangent lengths are the same. Take the circle with center O through any one of the points of tangency. The circle will automatically go through all the points of tangency and be orthogonal to all 3 circles.

Note: The standard mistake is to be able to construct the center but then not construct the radius.

Compare with Sved, Chapter 1, Problem 2 -- answer in the back of the book.

Problem 7.2: Construction of Orthogonal circles using tangent lines

Construct each of the following in an example.

1. Given a circle c and points P and Q on c, construct the circle p through P and Q that is orthogonal to c.

The center O of the circle is the intersection of the tangent lines to c at P and Q. The circle with center O through P will also pass through Q. See GTC, Chapter 9.

1. 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.

The center O of the circle is the intersection of the tangent line to c at Q and the perpendicular bisector of PQ. The circle with center O through P will also pass through Q. See GTC, Chapter 9.

Problem 7.3: Construction of Orthogonal circles using inversion

Construct each of the following in an example.

1. 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.

A circle passes through P and P' if and only if it is orthogonal to c. Likewise for P and P'' and d, so a circle through all 3 points must be orthogonal to both c and d.

1. 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 q.

A circle passes through P and P' if and only if it is orthogonal to c. Therefore, the circle q through P, P' and Q is orthogonal to c. But for the same reason, then the circle must pass through Q'. Finally, the line OQ must contain Q' and also q must contain Q', so Q' is a point of intersection of OQ and q. Q is one of these points, so Q' must be the other.

Problem 7.4: Construction of Images of Orthogonal circles by inversion

Construct each of the following separate figures (large, careful figures).  In each case, first draw a circle c with center O and radius r.

Figure 1. Draw a line m not intersecting c and construct the inversion image of m.  Construct a line n tangent to c and perpendicular to m.  Construct the inversion image of n.

Figure 2.  Construct an equilateral triangle inscribed in c.  Construct the inversion image of this triangle.

Figure 3.  Draw a circle e exterior to c, with O exterior to e. Construct the inversion image of this circle e.  Construct a circle f that intersects c at two points and with O interior to f. Construct the inversion image of this circle f.

Problem 7.5: More Images of circles by inversion

Construct two "random" circles a1 and a2 intersecting in two points A and B. Construct two circles b1 and b2 orthogonal to a1 and a2.

Now construct the inversion of this figure in a circle c with center A that passes through B.  (You can use Sketchpad.)

Explain in words how you can predict the figure that the inversion will look like.  Explain what circles are the inversions of all the circles a through A and B.  Explain what circles are the inversions of all the circles b that are orthogonal to a1 and a2.  Now use the word "pencil" in explaining what is happening.

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