431- 
GEOMETRY. 
extreme, be in the ratio of equality', and either mean, and the 
ratio nf the produel of the' extremes to the other mean, will be 
alfo iH a ratio of equality. —I'liat is, if A : B :: C : D; 
Then And 
U C 
Since A : B : C : D, by fnppofition; 
Therefore AxD=:BxC (162). 
. , AxD BXC , ^ BxC AxD 
And (157). Alfo-g-=-^ (157). 
^ X C 
But B ~—:— (i49)« 
Tlierefore 
C 
AxD_ 
C 
:B, 
^ ^ AxD ^ ^ 
And A= —^ (149)- 
A j BxC 
And—^=A (46), 
165. In any plane triangle, A B C, any two adjoining fdes, 
g p a AB, AC, are cut proportionally by 
a line D E, drawn parallel to the other 
fide'QC,viz. AD: DB :: AE : EC. 
Tlirough B and C draw Bb, Ca, 
at right angles to B C, meeting 
ba, drawn through A, parallel to 
B C : Througli p, q, tlie middles 
ot Ab, A a, draw pp, qq, parallel 
to B i or C:7, meeting A B, AC, 
in the points d, c, and join dc. 
Now the triangle A dp, '&dp, 
and Acq, Ccq, are congruous. 
^ heretore AB AczrzQc, pd~pd, qc—qc. 
h-ut p p q q (116). 'I'lierefore^ c. 
And dc \ % parallel to B C. 
In the fame manner it may be fliewiv, that lines parallel 
to B b, drawn through the middles oi Ap, pb- Aq, qa-, 
V. ill alio bifedf A:/, B ; Ac, Cc; and that 1 ines joining 
thefe points of biledtion will alfo be parallel to B C. 
And the fame may be proved at any other bifedlions of 
the fegments of the lines A B, A C.' Alfo the like may 
be readily inferred at any other divilions of the lines 
Ab, A a. Therefore lines parallel to B C^ cut off like 
parts from the lines A B, AC. 
Then AB: AC:: AD; AE. And AB:AC:BD:CE (ici). 
Therefore A D : AE ; B D : C E (1^5). 
And by alternation, A D : B D :: A E :' C E (145). 
Cor. Hetice, when the Tides AB, A C, of a triangle are 
cut proportionally in D, E, the fegments AD, AE; 
DB, EC; of th.ofe fides, are proportional to the fides. 
And the line DE, drawn to thofe fedtions, is parallel to 
the other fide B C. 
166. If feveral right lines meeting, or 
interfeBing each other in a point P, are cut 
by. two parallel lines A B, C D ; the inter¬ 
cepted fegments will be refpeElively propor¬ 
tional. 
AG : C O :: G N : O I:: N B : I D, &c. 
.A. G 
AT . B 
For the triangles APG, CPO; 
GPN, OPI ; NPB, IPD, are re- 
fpedtivcly equi-angular, and there¬ 
fore fimilar; hence AG : CO 
GP : OP ;: GN ; 01 ;: NP : IP :: 
NB : ID, &c. therefore AG ; CO 
:: GN ; 01 ;; NB ; ID, &c. 
AG A B 
Corol. Hence It is evident, if 
AG, GNj- See. and GO, OI, &c. 
are not in the fame continued right 
lines, but refpedtively parallel as 
. before, that CO, OI,.ID, &c. will 
B be in the fame proportion as AG, 
' GN, NB, 4 :c. 
167. In equiangular triangles, ABC, abc, the fides about 
the equal angles are proportional ; and 
the fides oppofitc- to equal angles are 
afo proportional. —In CA, CB, 
take CD—ca, C.¥.—cb‘, and draw 
DE ; then the triangles CDE, 
cab, being congruous (99), the 
Z,CDlL—{/_az=.)/_A. Therefore 
DE is parallel to AB (95). In 
tlie fame manner, taking AV—ac, AG .1 
AG=:a^; alfo BHrzz^c, BI—ia; and drawing EG, HI, 
the triangles .A-GF, IBH, ahe, are congruous; there, 
fore FG is parallel to CB, and HI is parallel to CA. 
Then (CD=) ca : CA ; : (CD=) cb : CB. 
(AF=r:) ca 
(BH=) be 
CA 
BC 
(AG; 
(Ble: 
;) ab 
) ab 
AB. 
AB. 
i, the ZA ^ 
right an- 
168. Cor. Hence, triangles having one angle in each 
equal, and the fides about thofe equal angles propor¬ 
tional, thofe triangles are equiangular and fimilar. 
169. In a right-angled triangle, ABC, if a line BD, be 
drawn from the right angle H, 
perpendicular to the oppofue fcle, —..B 
AC ; then will the triangles ABD, 
BCD, an each fide the perpendi¬ 
cular, be fimilar to the whole ABC, 
and to one another. —For in the 
triangles ABC, ADB, 
is common ; and the ^ 
gle ABC = right angle ADB. Therefore the remain- 
ing zC=ZABD. In the fame manner it will appear, 
that tiie triangles ABC, BDC, are like; therefore the 
triangles ABD, BCD, are alfo fimilar. 
170. Cor. I. Hence, AC ; AB : : AB ; AD. 
AC : BC : : BC : DC. 
AD : DB : : DB : DC. 
Cor. 2. Hence a right line B D, drawn from a 
circumference of a circle perpendicular to the diameter 
AC, is a mean proportional between the fegments AD, 
DC, of the diameter; and ADxDC^iDB^; for a 
circle, the diameter of which is AC, will pal's through 
A, B, C. (131.) 
Scholium. This corollary includes what is ufually 
called one of. the chief properties of the circle, namely : 
■—The fquare of the ordinate is equal to the recdangle 
under the two afafeiffas.—Here, the ordinate is the per¬ 
pendicular BD; and the two abfeiffes are the two feg- 
me.nts AD, DC, of the diameter AC. 
171. To make a fquare equal to a given 
reQangle ABCD.—Extend AB till 
j. BR=zBC, and on AR deferibe a fe- 
iR micircle; then produce CB to G; 
and the fquare on BG will be equal 
to thereilangie under AB, BR, or BC. 
172. In a circle, if two chords, 
A B, CD, intefetl each other 
in E, either within the circle, or 
without, by prolonging them ; then 
the reClangle under the fegments, ter¬ 
minated by the circumference and 
their inierfcEtion, will be equal .— 
That is, AExEBz= CE X ED. 
Draw the lines BC, DA ; then 
the triangles DEA, BEC, are 
fimilar ; for the angle at E is 
equal (93), or common ;. and 
the as fianding on 
the fame arc AC (129); then 
the other angles are equal (98). 
Tlierefore AE : CE : : ED : EB B 
(167); confequently AEXEB= 
CEXED (162}- 
173 - f 
