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30.
SUPPLEMENT.

PROBLEM I.

HAVING two points given A and B, to determine the Locus of the cen-

ters of all ſuch circles as may be drawn through A and B.

Join
AB and biſect it in the point C, and through C, draw a perpendicular

to it CE, and continue it both ways in infinitum, and it is evident that this line

CE will be the Locus required.

## 31. PROBLEM II.

Having
two right lines given AB and CD, to determine the Locus of the

centers of all ſuch circles as may be drawn touching both the ſaid lines.

Case
1ft. Suppoſe AB and CD to be parallel; then drawing GI perpendicu-

lar to them both; biſect it in H, and through H draw EHF parallel to the two

given lines, and it will be the Locus required.

Case
2d. Suppoſe the given lines being produced meet each other in E,

then biſecting the angle BED by the line EHF, this line EHF will be the Locus

required.

## 32. PROBLEM III.

Having
two circles given whoſe centers are A and B; to determine the Locus

of the centers of all ſuch circles as ſhall touch the two given ones.

Cases
1ſt and 2d. Suppoſe it be required that the circles be touched outwardly

by both the given ones; or that they be touched inwardly by both the given

ones.

Then
joining the centers A and B, let AB cut the circumferences in C and

D and produced in P and O: let CD which is intercepted between the convex

circumſerences be biſected in E: ſet off from E towards B the center of the