and hence the angle FEG is equal to EGH, but FEG is a right one by Con-

ſtruction. Let now HI be drawn from H perpendicular to AB: then the two

triangles EHI and EHG having two angles in one HEI and EIH reſpectively

equal to two angles in the other HEG and EGH, and alſo the ſide EH com-

mon, by Euc. I. 26. HI will be equal to HG, and therefore the circle will

touch alſo the other line AB: and HG or HI equals the given line Z, becauſe

EF was made equal to Z, and HG and EF are oppoſite ſides of a paral-

lelogram.

##
9.
PROBLEM III.

Having
two circles given whoſe centers are A and B, it is required to draw

another, whoſe Radius ſhall be equal to a given line Z, which ſhall alſo touch

the two given ones.

This
Problem has various Caſes, according to the various poſition of the

given circles, and the various manner of deſcribing the circle required: but there

are ſix principal ones, and to the conditions of theſe all the reſt are ſubject.

Case
1ſt. Let the circle to be deſcribed be required to be touched outwardly

by the given circles.

Limitation
. Then it is neceſſary that 2Z, or the given Diameter, ſhould

not be leſs than the ſegment of the line joining the centers of the given circles

which is intercepted between their convex circumferences, viz. not leſs than CD

in the Figure belonging to Caſe 1ſt.

Case
2d. Let the circle to be deſcribed be required to be touched inwardly by

the given circles.

Limitation
. Then it’s Diameter muſt not be given leſs than the right line,

which drawn through the centers of the given circles, is contained between their

concave circumferences; viz. not leſs than CD.

Case
3d. Let the circle to be deſcribed be required to be touched outwardly

by one of the given circles, and inwardly by the other.

Limitation
. Then it’s Diameter muſt not be given leſs than the ſegment

of the right line, joining the centers of the given circles, which is intercepted

between the convex circumference of one and the concave circumference of the

other; viz. not leſs than CD.

Case
4th. Let one of the given circles include the other, and let it be re-

quired that the circle to be deſcribed be touched outwardly by them both.

Limitation
. Then it’s Diameter muſt not be given greater than the greater

ſegment of the right line, joining the centers of the given circles, which is in-

tercepted between the concave circumference of one and the convex circumference

of the other; nor leſs than the leſſer ſegment; viz. not greater than CD, nor

leſs than MN.