Construct a circle through E orthogonal to the two circles. (Note: A and B are the centers of the circles.
Answer:
The class did very well on this problem. Two methods were used. One was to invert E in circle with center A to get E' and to invert in circle with center B to get E''. Then the desired circle is the circle through the 3 points E, E', E''. The center of this circle is the intersection of any 2 of the perpendicular bisectors of these 3 points.
The second method was to construct the radical axis of the two circles by the "shortcut" of intersecting with random circles. Then the center of the desired circle was the intersection of this radical axis with any of the perpendicular bisectors of E, E', E''.
One practical point was that the answers were more accurate if they used the
radical axis and the perpendicular bisector of EE' than those that used the
bisector of EE''. The reason is that the angle between the latter two was much
smaller, so the drawing errors are magnified. If you were doing serious drafting,
this would be an important point. (Of course nothing was taken off for this
phenomenon.)
(a)Define Power of a Point A with respect to a circle c with center O and radius r.
Answer:
The Power of Point A with respect to circle c = |OA|2 - r2.
Note 1: Any alternate definition such as tangent length squared or the product of two secant lengths has two problems: (1) those definitions do not cover the case when the power is negative and (2) those were not the official definition of the course, so the other definitions change what needes to be proved and what is already known.
Note 2: Definitions are very specific, minimalist statements. When writing a definition, you should write the definition and not add on extra stuff you know which is not part of the definition. If you want to add that, OK, but separate it and label it as additional information, so that it does not appear to be part of the defintion.
(b) Define the radical axis of two circles.
Answer:The radical axis of two circles is the set of points A at which the power of the two circles at A are equal.
Note 3: The definition as the locus of centers of orthogonal circles is an important fact about the radical axis, but it is flawed as a definition because it does not include the points where the power is 0 or negative. So it is not completely equivalent to the defintion in terms of the powers.
(c) In the figure, C is the center of c and D is the center of d. Construct these circles (or as much as will fit on the page):
Answer:
Construct a tangent segment PT from point P to one of the circles as we did in 444 (for example, construct circle e with diameter PC and then intersect e with c to get a point T). Then draw the circle with center P and radius PT. Any of the 2 tangents from P to c or P to d will give the same circle. This is true because P is on the radical axis of circles c and d.
Answer:
There is no such circle with center Q because Q is not on the radical axis. If you try constructing tangent segments as for P, you get different lengths for c and for d, since Q is not on the radical axis.
