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p 215
Pro. 21
313
Upon
a given
strait
line
F.G,
to make
a figure
similar
to a
given
figure
ABC
DE
Join
AC,
AD,
make
the
angle
FGH
equal
to A
BC,
FH
G equal
to A
CD,
HI
J equal
to G
AD
by
problem 9
and
the
thing
is done
p 216
Pro. 22
314
To
make
a figure
similar
to
a given
figure
AB
CD, &
in the
given
ratio
to it of
M to
N.
Produce
BA to
E, find
AE a
fourth
proportional
to M,
N, and
BA,
by problem
15, upon
the
diameter
EB
describe
the
semicircle
EFB,
draw
AE at
right
angles
to EB,
make
AG
equal
to A
F, and
the
figure
AGH
I similar
to A
BGI,
by
problem
21.
p 217
Pro. 23
315
Upon
a given
strait
line
AB,
to describe
the
segment
of a
circle,
that
shall
contain
an
angle
equal
to a
given
angle
C.
Make
the
angle
BAD
equal
to the
given
angle
C, draw
AE at
right
angles
to AD,
bisect
AB
in F,
draw
FE,
perpendicular
to A
B, and
about
the
center
E, with
the
radius
E
A or
EB,
describe
the
segment
AG
B.
+
p 219
Pro. 24
316
The
base
of a
triangle,
the
verticle
angle,
and
the
ratio
of the
sides,
are given —
to make
the triangle
Make
the
Angle
ACB
equal
to the
given
angle,
and
CD
to CE
in the
given
ratio
of m
to n,
join
ED,
and
continue
it
to F,
draw
FA, parallel
to CB,
and
AB
parallel
to
FE.
p 220
Pro. 25
317
The angles
at the
base
of a
triangle,
&
a line
joining
the
vertex
and
middle
of the
base,
are
given
— to
make
the
triangle
In the
strait
line
AB
assume
DE,
DF
equal
to each
other
make
the
angles
FEG,
EFG
equal
to the
given
angles
at
the
base
each
to each,
join
DG
and
continue
it to
C,
make
D.C equal
to the
given
line,
and
draw
CA,
CB,
parallel
to
GE,
GF.
p. 221
Pro. 26
318
The
verticle
angle,
a line
joining
the
vertex
and
middle
of the
base,
and
the
angle
made
by the
said
line
and
base,
are
given —
to
make
the
rectangle.
In
the
strait
line
AB
take
DE,
DF equal
to each
other,
make
the angle
B
DC, equal
to the
given
angle
and D
C equal
to the
given
line,
upon
EF,
describe
the segment
of a
circle
that
will
contain
the vertical
angle,
by problem
13, cutting
DC in
G, join
EG, D
F, and
draw
GA, G
B, parallel
Pro 26
conclu.
to G
E, G
F
p 222
Pro 27
319
The
base
of a
triangle
AB,
the altitude
AD,
and
the ratio
of
its sides
AC, C
B are
given
as M
to N,
to make
the triangle
Divide
the base
in the
given
ratio
of the
sides
at E,
by problem
13, make
EF
equal
to EB,
produce
AB
to G,
find
EG as
fourth
proportional
to AF,
FE, A
E, by
problem
15, about
the
center
G, with
the radius
GE,
describe
the
circle
EGH,
draw
DG
parallel
to A
G, and
join
AC,
CB.
p 223
Pro 28
320
The
angles
at the
base
of a
triangle,
and
the
sum
of its
three
sides
are
given
— to
make
the
triangle
Let
DE
be the
given
sum
of the
sides;
make
the
angles
DEC,
EDC,
equal
to half
the
given
angles
at the
base
respectively,
by
problems
48, 9
make
the
angle
DCA,
equal
to CD
E, and
the
angle
ECB
equal
to C
ED.
p 224
Pro 29
321
To
inscribe
a triangle
ABC,
in a
given
triangle
DEF,
and
make
it equiangular
to another
given
triangle
G
HI.
Make
the
angles
FDK,
DEK
equal
to the
acute
angles
G and
H, each
to each,
join
KE
cutting
DF in
C, draw
CA
parallel
to K
D, CB
parallel
to K
F and
join
AB.
p 225
Pro. 30
322
To
inscribe
a triangle
in
a square
A
BCD
and
make
it equiangular
to a
given
triangle
E
FG.
Make
the
angles
BA
H, A
BH
equal
to the
two
least
angles
of
the
given
triangle
E & F,
each
to each
draw
HI
parallel
to
AD,
IK
parallel
to H
A, IL
parallel
to H
B, and
join
KL.
p 226
Pro 31
323
To
inscribe
a square
in a
given
triangle
ABC
Let
fall
the
perpendicular
C
D, draw
CE
parallel
to A
B, make
CE
equal
to CD,
join
AE
cutting
BC
in F,
draw
FG
parallel
to B
A, &
GH,
FI,
parallel
to C
D.
p 227
Pro 32
324
To
divide
a given
strait
line
AB,
so
that
the
rectangle
contained
by the
whole
line
& one
of the
parts,
shall
be equal
to the
square
of the
other
part.
Produce
BA
to C,
draw
AD
at
right
angles
to A
B, make
both
AC
& AD
equal
to AB
bisect
AC
in E
join
ED
make
EF
equal
to E
D and
F is
the
point
required.
p 228
Pro. 33
325
To
find
the
center
of a
given
circle
AB
CD.
Draw
EF at
pleasure
cutting
the
circle
in E
& F, bisect
EF at
right
angles
with
the
strait
line
BD,
by
problem
5; again
bisect
BD
at
right
angles
by the
line
AC
in G,
and
the
point
G will
be the
center
of the
circle.
p. 229
Pro. 34
326
From
a given
point
A, to
draw
a tangent
to a
given
circle.
Case I
When
the
given
point
is in
the
circumference
Find
E the
center
of
the
circle
by
problem
33,
join
EA
and
through
the
point
A,
draw
FG at
right
angles
to E
A by
problem
6.
+
+
p 232
Pro. 35
327
To
make
the
greatest
triangle,
that
shall
have
two
given
lines
AB,
BC,
for
two
of its
sides
Place
the
given
sides
AB, B
C at
right
angles
to
each
other,
by
problem
6, and
join
AC
p 233
Pro. 36
328
To
make
the
greatest
triangle
upon
a given
base
AB,
with
a given
vertical
angle
A
CB.
Upon
the
base
AB,
describe
the
segment
of a
circle
ABC
that
will
contain
the
given
angle
by
problem
23, bisect
AB
in D,
draw
DC
at
right
angles
to AB,
and
join
AC,
CB.
p 234
Pro 37
329
To
make
the
greatest
triangle
upon
a given
base
AB,
with
a given
line
AD,
for
the
sum
of its
sides
AC,
CB.
By
problem
8, make
an isosceles
triangle
ABC
whose
sides
AC,
BC
are
each
equal
to half
AD
and
the
thing
is
done
p. 235
Pro. 38
330
To
circumscribe
a circle
about
a given
triangle
ABC.
Bisect
any
two
sides
AB,
BC
in D
and
E,
draw
DF,
EF
at
right
angles
to
AB,
BC,
and
about
the
center
F, with
the
radius
F
A describe
the
circle
p 236
Pro. 39
331
In
a given
circle
ABC,
to inscribe
a triangle
that
shall
be equiangular
to a
given
triangle
D
EF.
Draw
GH a
tangent
to the
circle
at the
point
H, by
problem
34,
make
the
angle
HBC
equal
to ED
F, the
angle
GBA
equal
to DF
E, and
join
AC.
+
p 238
Pro 40
332
To
describe
triangle
about
a given
circle
ABC,
and
make
it equiangular
to a
given
triangle
DEF.
Produce
the
base
DE
both
ways
to G
& H,
find
I the
center
of
the circle,
join
IA,
make
the
angles
AIC,
AIB
equal
to G
DF,
FEH
each
to
each
and
thro'
the
points
A, B, C
draw
KL,
LM,
KM
tangents
to the
circle.
p 239
Pro. 41
333
To
inscribe
a circle
in a
given
triangle
ABC
Bisect
any
two of
the
angles
ABC,
CAB,
by the
lines
ED,
AD,
draw
DE,
perpendicular
to A
B, &
about
the
center
D
with
the
radius
DE
describe
the
circle.
p 240
Pro 42
334
To
inscribe
a
square
in a
given
circle
AB
CD
Draw
the
diameters
AC,
BD
at
right
angles
to
each
other,
and
join
AB,
BC,
CD,
DA.
p 241
Pro. 43
335
To
inscribe
a hexagon
in a
given
circle
AB
CD
&c
Find
G the
center
of
the
circle,
draw
the
diameter
AD,
about
the
center
A,
with
the
radius
AG,
describe
a circle
B
GF,
join
BG,
FG,
produce
them
to E
& C,
and
draw
AB,
BC,
CD,
DE,
EF,
FA.
p 242
Pro. 44
336
To
inscribe
a
regular
pentagon
or
decagon,
in a
given
circle.
Let
AB
CD
&c be
the
inscribed
decagon,
join
AC,
AD,
find
L the
center
of
the
circle
and
draw
BL,
LC,
LD.
By
problem
31
draw
the
diameter
AF
and
the
radius
LN
at
right
angles
to it,
bisect
AL
in O
make
OP
equal
to ON
and
join
NP.
Then
LP =
(ML -
CD
Prob 44
conclu.
is the
side
of the
decagon
and
NP =
AC, is
the
side
of the
pentagon,
which
must
be applied
the
p 244
Pro. 45
337
About
a given
circle
to describe
a regular
figure
having
the
same
number
of sides
with
a regular
figure
inscribed
in
that
circle.
Let A,
B, C, D,
E, be
the
angular
points
of the
inscribed
figure
and
at the
points
A, B,
C, D,
E, by
problem
34,
draw
tangents
to the
circle,
meeting
one another
at F, G,
H, I, K.
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geometry iii |
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