Two circles have radii 1 and 3 and centers a distance 10 apart. Find the locus of all points which are the midpoint of a segment with one end on each circle.

**Solution**

Answer: Let O_{1}, O_{2} be the centers of C_{1}, C_{2} and O the midpoint of O_{1}O_{2}. The locus is the annulus radii 1 and 2, center O.

*Easy.*

Fix Y on the C_{2}. Then the locus of the midpoint is a circle radius 1/2, center N, the midpoint of XO_{1}. Now vary X. The locus of N is a circle radius 1 1/2 center O. Hence the locus of M is the area swept out by the circle radius 1/2 as its center moves around the circle radius 1 1/2. This is an annulus radii 1 and 2.

© John Scholes

jscholes@kalva.demon.co.uk

12 Dec 1998