SS-l THEORY OF
AIICIIITECTUIIE.
Book II.
will also (Prop. 84.) he
equiangular to ahc
;
which is imptssil)le, since the triangle ABC
is suj>posefl equiangular to <ibc.
<)G.K Pitop. EX XX VI. If four lines are proportional, their srjuarc.s are a/so proportional.
If the line A B he to the line AC as the line A D is to the line AF
(.
31
y,)>
^ln-' square
of the line AB will he to the scjuare of the line A(^
as the square of the line AD is to the square of the
line AF.
With the lines AB and AD form an angle BAD;
with the lines AC and AF form another angle CAF
e(]ual to the angle BAD, and draw the right lines
BD, CF.
Because AB is to AC as AD to AF, and the con-
"
tained angles are equal, the two triangles BAD, CAF
have their sides ahout e(jual angles proportional ; they are therefore (Prop. 63.)
cquir.ngular,
and consequently (Prop. 85.)
similar : whence they are to one another (Prop. 8'i.) as
the squares of their homologous sides. If, then, the triangle BAD he a third ))art of
the triangle CAl*", the square of the side AB will be a third part of the square of the side
AC, and the square of the side AD will be a third part of the square of the side A F.
Wherefore these four squares will be proportional.
966. Pkop. LXXXVII. Similar rectiliueal Jiynres may be divick<l into an equal numher
of
simi!ar triangles.
Let the similar figures he A BC DF, a^ciT/", and draw the homologous lines CA,fa, CF,(/;
these two figures will be divided into an equal number of
similar triangles.
The triangles BCA, hca
(fff.
320.),
being composed of an
^
,
etpial number of corresponding points, are similar. The / / \^
^D
/j^k
triangles ACF,
Of/ and the triangles F'CD,/f(/ are also, for
i^x
/ \ /
j,
/
' )**
the same reason, similar. Wherefore the similar figures
\l
\l \l
'''.
A BCDF, a&cf//" are divided into an equal number of similar
A t
J
triangles.
967. Prop. LXXXVIII. Similar
figures
are equiangular.
The similar figures ABCDF", afccf/f (see
fig.
preced. Prop.) have their angles e(|ual.
Draw the homologous lines CA, ca, CF,
cf.
The triangles BCA, bca are similar, and ton-
setniently
equiangular. Therefore the angle B is e<|ual to the angle b, the angle BAt' to
the angle bac, and the angle BCA to the angle bea. The triangles ACF, ucf, l'CD,/(7i
are also equiangular, because they are similar. Therefore all the angles of tl;e similar
figures ABCDF', abcdf are equal.
968. Prop. LXXXIX. Equiangular
figures
the sides
of
which are proportional are
similar.
If the figures A BCDF,
aicr?/"
{fig.
3'21.) have their angles equal and their sides propor-
tional, they are similar. Draw the right lines CA, oa,
c
CF,cf.
The triangles CBA, c&a, have two sides proportional and
the
contained angle equal
;
they are therefore (Prop. G'^.)
equiangular, and consequently (Prop. 85.)
similar. The
lines CA, ca are therefore (Prop.
80.)
proportional.
The triangles CAF, c/ have two sides proportional and
the
contained angle equal; for if from the equal angles
Fis.Mi.
BAF, i/'be taken the equal angles BAC, ?)nc, there will remain the equal angles CAF,
caf.
These two triangles are therefore equiangular, and consequently similar. In the
same
manner it may be proved that the triangles CFT), cfd are similar.
The two figures ABCDF, abcdfaxe then composed of an equal number of similar triangles;
that is, they are composed of an equal number of points disposed in the same manner, or
are similar.
969.
DEFiNrnoNs.
\. A plane is a surface, such that if a right line
ai)i)lied to if
touch it in two points it will touch it in every other ])()int.
The surface of a fluid at rest, or of a well-polished table, m^y
be considered as a jjlane.
2. A right line is perpendicular to a jjlane if it make right
angles with all lines which can be drawn from any point in
that plane. Thus BA
{fig.
322.) is jierpendicular to the i)lane
^^^
MI.iGFl'N, because it makes right angles with the lines AM,
m^'
-p
A L, AG, &c. drawn from the jioint .\.
~"n~' 'V
S Lei AH
(Jig.
323.) be the common intersection of two planes.
Fit;.3ia.
