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Let A and B be the centres of the given circles.
It is required to draw their radical axis.

If the given circles intersect, then the st. line drawn through their points of intersection will be the radical axis. [Ex. 1, Cor. p. vi.]

But if the given circles do not intersect,

describe any circle so as to cut them in E, F and G, H.
Join EF and HG, and produce them to meet in P.

Join AB; and from P draw PS perp. to AB.

Then PS shall be the radical axis of the Os (A), (B).

DEFINITION. If each pair of circles in a given system have the same radical axis, the circles are said to be co-axal.

EXAMPLES.

1. Shew that the radical axis of two circles bisects any one of their common tangents.

2. If tangents are drawn to two circles from any point on their radical axis; shew that a circle described with this point as centre and any one of the tangents as radius, cuts both the given circles orthogonally.

3. O is the radical centre of three circles, and from a tangent OT is drawn to any one of them: shew that a circle whose centre is O and radius OT cuts all the given circles orthogonally.

4. If three circles touch one another, taken two and two, shew that their common tangents at the points of contact are concurrent.

5. If circles are described on the three sides of a triangle as diameter, their radical centre is the orthocentre of the triangle.

6. All circles which pass through a fixed point and cut a given circle orthogonally, pass through a second fixed point.

7. Find the locus of the centres of all circles which pass through a given point and cut a given circle orthogonally.

8. Describe a circle to pass through a given point and cut two given circles orthogonally.

9. Find the locus of the centres of all circles which cut two given circles orthogonally.

10. Describe a circle to pass through a given point and cut two given circles orthogonally.

11. The difference of the squares on the tangents drawn from any point to two circles is equal to twice the rectangle contained by the straight line joining their centres and the perpendicular from the given point on their radical axis.

12. In a system of co-axal circles which do not intersect, any point is taken on the radical axis; shew that a circle described from this point as centre with radius equal to the tangent drawn from it to any one of the circles, will meet the line of centres in two fixed points.

[These fixed points are called the Limiting Points of the system.].

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