• Prove that the radical axis is a line. (Note that the word "line" is not in our definition of radical axis.
• Prove that for 3 circles, the 3 radical axes of pairs of the circles are either concurrent or parallel (with special case for two circles concentric).
• Prove that if c is a circle and A and B are distinct points so that the inversion of A in c is B, then any circle or line through A and B is orthogonal to c.
• Prove that the inversion image of any line is either a circle or a line (explain cases).
• Harder: Prove that the inversion image of any circle is a circle or a line.

• If c is a circle and A is a point not on the circle (and not the center of the circle), let A' be the inversion of A in c. Then prove that any circle d though A and A' is orthogonal to c.
• Given circles c1 and c2 that intersect in points A and B and also a circle d orthogonal to c1 and c2. Prove that if c3 is another circle passing through A and B, then d is also orthogonal to c3.
• What is the definition of the radical axis of two circles? (Note: the word "line" does not appear in this.)
• Prove that the radical axis is a line.
• If two circles intersect, why does the radical axis pass through the points of intersection?
• Suppose that two circles c1 and c2 have a common tangent ST, then show the radical axis of the two circles passes through the midpoint of ST.
• In lab we also definted the radical axis of a circle C1 and a point P. Explain why this is the usual definition in terms of powers of a point if you allow a "circle" to have radius 0.
• Let m be the radical axis of circle c1 and point P. If O is on m, explain why circle d with center O orthogonal to c1 must pass through P.
• Given two circles c1 and c2 and a point P, prove that one can construct a circle with center P that is orthogonal to both the given circles only when P is on the radical axis. Is it possible to have a point P on the radical axis of two circles but there does not exist a circle with center P which is orthogonal to the two circles? [Hint: Consider two intersecting circles. What happens to the power of P when there is no circle?]
• If a circle d is known to be orthogonal to circle c1 and the center of d is on the radical axis of circles c1 and c2, then explain why d is also orthogonal to c2.
• Suppose a figure consists of two orthogonal circles c and d (and their centers). For any point A on d, tell how one can construct with a straightedge only the inversion of A in c. What is the inversion of d in c?

Study note: Some of these questions are posed in Sved, Chapter One, e.g., problems #2, #4, #9. There are answers for the Sved problems in the back of the book.

Study Constructions

• Given two circles (or a point and a circle), construct the radical axis of the two circles (or the point and the circle).
• Given 3 circles, construct the radical center of the 3 circles.
• Figure out this set of four important related constructions: Given p circles and q points, find a circle through the q points that is orthogonal to each of the p circles. This generally works if p+q = 3, where p = 0, 1, 2, 3. Be sure to consider special cases when some of the points are on the circles, are interior to the circles, or exterior to the circles.

Study note: Some of these constructions are posed in Sved, Chapter One, e.g., problems #8, #10, #11, #12. There are answers for the Sved problems in the back of the book.