# Trapezium

##### 6 minutes • 1098 words

## Table of contents

## Lemma 27

A trapezium given in kind, the angles whereof may be in placed, respect of four right lines given by position, that are neither all paralhl among themselves, nor converge to one common point, describe
,
the several angles may touch the several lines.
Let the four right lines ABC, AD, BD, CE, be
given by position the first cutting the second in A,
the third in B, and the fourth in C and suppose a
;
trapezium fghi
is to
be described that
may
be similar
trapezium FCHI, and whose angle /, equal to
the given angle F, may touch the right line ABC and
to the
;
other angles g, h, i, equal to the other given angles,
G, H, I, may touch the other lines AD, BD, CE, re
Join FH, and upon FG. FH, FI describe
spectively.
(lie
as
of
J%
many segments of circles FSG, FTH, FVI, the first
which FSG may be capable of an angle equal to
BAD
FTH
the second
the angle
capable of an angle
the
third
and
FVI of an angle equal to the angle
to
the
equal
angle
the segments are to be described towards those sides of the
ACE.
;
CBD
;
Bnrf>,
lines
the
FG, FH,
same
FI, that the circular order of the letters
BADB, and that the letters
.ibout in the
same order
FSGF may
FTHF
as of the letters
as the letters
CBDC
and the
letters
may
be
turn
FVIF in the
game order as the letters ACE A. Complete the segments into entire cir
cles, and let P be the centre of the first circle FSG, Q, the centre of the
Join and produce both ways the line PQ,, and in it take
second FTH.
QR in the same ratio to PQ as BC has to AB. But QR is to be taken
towards that side of the point
Q
that the order of the letters P,
Q,,
ROF NATURAL PHILOSOPHY.
SEC. V.J
may
be the same as of the letters A, B, C
with the interval
;
R
and about the centre
RF
describe a fourth circle
third circle
(lie
FVI
in
FNc
Join
c.
cutting
Fc cut
1
and the second in
and let the figure
first circle in a,
ting the
Draw aG, &H,
be made
/ .
15]
ABC/ 4f/ii
cl,
similar to the figure
w^cFGHI; and the trapezium fghi will
be that which was required to be de
scribed.
For
let the
two
RK,
The
"K,
FSG, FTH
join PK, Q,K,
first circles
K
cut one the other in
;
QP
6K, cK, and produce
angles
FaK, F6K, FcK
to
L.
at the circumferences are the halves of the
FPK, FQK, FRK,
LRK, the halves of
at the centres, and therefore equal to LPK,
those angles.
Wherefore the figure
is
to the figure
and
and
similar
ab
is
be
to
6cK,
consequently
iquiangular
to BC.
But by construction, the angles
res PQ, to
Q,R, that is, as
angles
LQ.K,
PQRK
AB
/B//,/C? are equal
Air,
,
to the angles
FG,
F&H,
Fcl.
And
therefore
be completed similar to the figure abcFGHl.
vVliich done a trapezium fghi will be constructed similar to the trapezium
FGHI, and which by its angles/, g, h, i will touch the right lines ABC,
the figure
ABCfghi may
AD, BD, CE.
Q.E.F.
COR. Hence a right line may be drawn whose parts intercepted in a
given order, between four right lines given by position, shall have a given
Let the angles FGH, GHI, be so far in
proportion among themselves.
creased that the right lines FG, GH, HI, may lie in directum ; and by
constructing the Problem in this case, a right line fghi will be drawn,
whose parts fg, gh, hi, intercepted between the four right lines given
by
AD, AD and BD, BD and CE, will be one to another
lines FG, GH, HI, and will observe the same order among them
But the same thing may be more readily done in this manner.
AB
position,
as the
selves.
Produce
so as
GH
BK
LM
AB
may
to
K
BD to L,
AB as HI to
BD as GI to FG;
and
be to
DL to
KL meeting
and
;
and join
CE
and
the right line
Produce iL
to iL as
be
may
in
i.
to
M,
GH
II
so as
to
HI
;
draw MQ,
and
AD
and
M*
join gi cutting AB, BD in f, h I
then
parallel to LB,
in g,
meeting the right line
;
say, the thing is done.
For
let
MO- cut the right
line
AB
in
Q, and
AD
the right line
KL
iuTHE MATHEMATICAL PRINCIPLES
^52
to hi,
AP
draw
S, arid
Mi
to Li,
to
in 11,
DL
Cut
ratio.
parallel to
GI
AS
cause ffS to g~M,
(ex
ceqit.o)
gS
RL to
as AD is to
BD as
BL
Eh
to
as
Lh
D
BR
therefore
it is
and //..
BK)
to
AP. and
RL
DS
and
AP
may
to
in P, and -M to Lh (g

BL, will be in the same
to
be in that same ratio; and be
DL
are proportional; therefore
AS be to BL, and DS to RL and mixtly.
AS
DS to gS AS. That is, BR is to
LA, so will
BL,
as
;
Ag, and therefore as BD to gQ. And alternately BR is
But by construction the line BL
or as fh to fg.
and R in the same ratio as the line FI in G and H and
Wherefore fh is to fg as FH to
is to BD as FH to FG.
;
Since, therefore,
HI,
DL
to
I.
13/i to g-Q,,
was cut in
FG.
to
HI,
so as
to
BD, and meeting iL
AK
[BOOK
gi
to hi likewise is as
Mi
to Li, that
manifest that the lines FI, fi, are similarly cut in
GI
is, as
G and H,
to
g
Q.E.F.
LK is drawn cutting
Ei as FH to HI,
In the construction of this Corollary, after the line
in i, we may produce iE to V, so as
may be to
EV
CE
then draw V/~ parallel to BD. It will come to the same, if about the
in X, and
i with an interval IH, we describe a circle
cutting
so as iY may be equal to IF, and then draw Yf parallel
produce iX to
arid
BD
centre
Y
to
BO.
Sir Christopher Wren and Dr. Wallis have long ago given other solu
tions of this Problem.