99[22]
Analysis.
Let us conceive the thing effected, and that O is really the
point ſought. Then, by ſuppoſition, the rectangle AO, EO is to the
ſquare on IO as R to S. Make EC to IC as R is to S; and the rectangle
AO, EO is to the ſquare on IO as EC to IC. Let now OB be taken a
fourth proportional to EO, EC and IO; then (Eu. V. 15.) the rectangle
AO, EO is to the ſquare on IO as the rectangle EC, OB is to the rectangle
IC, OB; and ſo by permutation, the rectangle AO, EO is to the rectangle
EC, OB as the ſquare on IO is to the rectangle IC, OB; and becauſe EO is
to EC as IO to BO, AO will be to OB as IO to IC, and ſo by compoſition,
or diviſion CO is to EC as IB to OB, and AB is to OB as CO to IC;
whence, ex æquo perturb. et permut. AB is to IB as EC to IC; that is in the
given ratio, and hence is given BC, the ſum or difference of CO and BO,
as alſo the rectangle contained by them, equal to the rectangle AB, IC,
wherefore theſe lines themſelves are given by the 85th or 86th of the Data.
point ſought. Then, by ſuppoſition, the rectangle AO, EO is to the
ſquare on IO as R to S. Make EC to IC as R is to S; and the rectangle
AO, EO is to the ſquare on IO as EC to IC. Let now OB be taken a
fourth proportional to EO, EC and IO; then (Eu. V. 15.) the rectangle
AO, EO is to the ſquare on IO as the rectangle EC, OB is to the rectangle
IC, OB; and ſo by permutation, the rectangle AO, EO is to the rectangle
EC, OB as the ſquare on IO is to the rectangle IC, OB; and becauſe EO is
to EC as IO to BO, AO will be to OB as IO to IC, and ſo by compoſition,
or diviſion CO is to EC as IB to OB, and AB is to OB as CO to IC;
whence, ex æquo perturb. et permut. AB is to IB as EC to IC; that is in the
given ratio, and hence is given BC, the ſum or difference of CO and BO,
as alſo the rectangle contained by them, equal to the rectangle AB, IC,
wherefore theſe lines themſelves are given by the 85th or 86th of the Data.
Synthesis.
Make AB to IB and EC to IC in the given ratio, and
deſcribe on BC a circle; erect, at B, the indefinite perpendicular BK, and
take therein BD a mean proportional between AB and IC, or between IB
and EC: from D draw DH parallel to CB, if O muſt fall between B and C;
but through F, the center of the circle on BC, if it muſt fall without them,
cutting the ſ@id circle in H; then draw HO perpendicular to DH, which
will cut the indefinite line in O, the point required.
deſcribe on BC a circle; erect, at B, the indefinite perpendicular BK, and
take therein BD a mean proportional between AB and IC, or between IB
and EC: from D draw DH parallel to CB, if O muſt fall between B and C;
but through F, the center of the circle on BC, if it muſt fall without them,
cutting the ſ@id circle in H; then draw HO perpendicular to DH, which
will cut the indefinite line in O, the point required.
For it is plain from the conſtruction that BD and HO are equal, and
(Eu. IV. 17.) the rectangle AB, IC, or the rectangle IB, EC is equal to the
ſquare on BD, and therefore equal to the ſquare on HO, which (Eu. III.
35. 36.) is equal to the rectangle BO, CO: conſequently (Eu. VI. 16.) AB
is to BO as CO to IC; alſo EC is to CO as BO is to IB; wherefore, by
compoſition or diviſion, AO is to BO as IO to IC, and EO to EC as IO
to BO: conſequently by compound ratio, the rectangle contained by AO and
EO is to the rectangle contained by BO and EC, as the ſquare on IO is to
the rectangle contained by BO and IC; by permutation, the rectangle
contained by AO and EO is to the ſquare on IO as the rectangle contained
by BO and EC is to the rectangle contained by BO and IC, that is (Euc.
v. 15.) as EC is to IC, or as R to S.
(Eu. IV. 17.) the rectangle AB, IC, or the rectangle IB, EC is equal to the
ſquare on BD, and therefore equal to the ſquare on HO, which (Eu. III.
35. 36.) is equal to the rectangle BO, CO: conſequently (Eu. VI. 16.) AB
is to BO as CO to IC; alſo EC is to CO as BO is to IB; wherefore, by
compoſition or diviſion, AO is to BO as IO to IC, and EO to EC as IO
to BO: conſequently by compound ratio, the rectangle contained by AO and
EO is to the rectangle contained by BO and EC, as the ſquare on IO is to
the rectangle contained by BO and IC; by permutation, the rectangle
contained by AO and EO is to the ſquare on IO as the rectangle contained
by BO and EC is to the rectangle contained by BO and IC, that is (Euc.
v. 15.) as EC is to IC, or as R to S.
Q.
E.
D.
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