14(2)
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.