DC is drawn to the given point D in the ftraight line BD given c. 32. Dat. in pofition in the given angle BDC, DC is given in pofition. and d. 28. Dat. the circumference ABC is given in pofition, therefore the point C is given.

91.

IF

d

F from a given point a straight line be drawn touching a circle given in pofition; the ftraight line is given in pofition and magnitude.

Let the ftraight line AB be drawn from the given point A touching the circle BC given in position; AB is given in position and magnitude.

Take D the center of the circle, and join DA, DE. because each of the points D, A is given, the ftraight

a

a. 29. Dat. line AD is given in pofition and magni- C

b. re. 3.

tude. and DBA is a right bangle, where

c. Cor. 5. 4. fore DA is a diameter of the circle DBA

defcribed about the triangle DBA; and

d. 6. Def. that circle is therefore given in pofition.

and the circle BC is given in pofition, there

e. 28. Dat. fore the point B is given. the point A is alfo given; therefore the ftraight line AB is given in pofition and magnitude.

92.

a. 17.3.

a

IF a ftraight line be drawn from a given point without a circle given in pofition; the rectangle contained by the fegments betwixt the point and the circumference of the circle is given.

Let the ftraight line ABC be drawn from the given point A without the circle ECD given in pofition,

cutting it in B, C; the rectangle BA, AC
is given.

From the point A draw AD touching c b. 94. Dat the circle; therefore AD is given b in pofition and magnitude. and because AD is

c

c. 56. Dat. given, the fquare of AD is given which

d. 36. 3.

A

is equal to the rectangle BA, AC. therefore the rectangle BA, AC is given.

IF

PROP. XCVI.

F a ftraight line be drawn thro' a given point within a circle given in pofition, the rectangle contained by the fegments betwixt the point and the circumference of the circle is given.

Let the ftraight line BAC be drawn thro' the given point A within the circle BCE given in pofition; the rectangle BA, AC is given. Take D the center of the circle, join AD

93.

and produce it to the points E, F. becaufe

the points A, D are given, the ftraight line

AD is given in pofition; and the circle BEC

E

C

A

b. 28. Dat.

is given in pofition; therefore the points E, B F are given . and the point A is given, F therefore EA, AF are each of them given"; and the rectangle LA, AF is therefore given; and it is equal to c. 35. 3. the rectangle BA, AC which confequently is given.

IF

PROP. XCVII.

C

F a ftraight line be drawn within a circle given in magnitude cutting off a fegment containing a given angle; if the angle in the fegment be bifected by a ftraight line produced till it meets the circumference, the ftraight lines which contain the given angle fhall both of them together have a given ratio to the ftraight line which bifects the angle. and the rectangle contained by both thefe lines together which contain the given angle, and the part of the bifecting line cut off below the base of the fegment, fhall be given.

F

Let the ftraight line BC be drawn within the circle ABC given in magnitude cutting off a fegment containing the given angle BAC. and let the angle BAC be bifected by the ftraight line AD; BA together with AC has a given ratio to AD; and the rectangle contained by BA and AC together, and the ftraight line ED cut off from AD below BC the bafe of the fegment, is given.

I

E

D

94.

Join BD; and because BC is drawn within the circle ABC given in magnitude cutting off the fegment BAC containing the given a. 91. Dat. angle BAC; BC is given in magnitude. by the fame reafon BD b. 1. Dat. is given; therefore the ratio of BC to BD is given. and because the angle BAC is bifected by AD, as BA to AC, fo is BE to EC; 2. 12. 5. and, by permutation, as AB to BE, fo is AC to CE; wherefore ◄ as BA and AC together to BC, fo is AC to CE. and because the

c. 3. 6.

8. 21. 3.

angle BAE is equal to EAC, and the F

c

angle ACE to ADB; the triangle ACE
is equiangular to the triangle ADB;
therefore as AC to CE, fo is AD to DB.
but as AC to CE, fo is BA together with
AC to BC; as therefore BA and AC to
BC, fo is AD to DB; and, by permu-

Α

tation, as BA and AC to AD, so is BC to BD. and the ratio of BC to BD is given, therefore the ratio of BA together with AC to AD is given.

Alfo the rectangle contained by BA and AC together, and DE is given.

Because the triangle BDE is equiangular to the triangle ACE, as BD to DE, fo is AC to CE; and as AC to CE, fo is BA and AC to BC; therefore as BA and AC to BC, fo is BD to DE. wherefore the rectangle contained by BA and AC together, and DE is equal to the rectangle CB, BD. but CB, BD) is given; therefore the rectangle contained by BA and AC together, and DE is given. Otherwife.

Produce CA and make AF equal to AB, and join BF. and bea. 5. and 32. cause the angle BAC is double of each of the angles BFA, BAD,

1.

the angle BFC is equal to BAD; and the angle BCA is equal to BDA, therefore the triangle FCB is equiangular to `ADB. as therefore FC to CB, fo is AD to DB, and, by permutation, as FC, that is BA and AC together to AD, fo is CB to BD. and the ratio of CB to BD is given, therefore the ratio of BA and AC to AD is given.

And because the angle BFC is equal to the angle DAC, that is to the angle DBC, and the angle ACB equal to the angle ADB; the triangle FCB is equiangular to BDE, as therefore FC to CB, fo is BD to DE; therefore the rectangle contained by FC, that is BA and AC together, and DE is equal to the rectangle CB, BD which is given, and therefore the rectangle contained by BA, AC together, and DE is given.

Fa fraight line be drawn within a circle given in magnitude cutting off a fegment containing a given angle; it the angle adjacent to the angle in the tegment be bifeded by a fraight line produced till it mect the circumference again and the bafe of the fegment; the excefs of the ftraight lines which contain the given angle fhall have a given ratio to the fegment of the bifecting line which is within the circle; and the rectangle contained by the fame excess and the fegment of the bifecting line betwixt the bafe produced and the point where it again meets the circumference, fhall be given.

Let the ftraight line BC be drawn within the circle ADC given in magnitude cutting off a fegment containing the given angle BAC, and let the angle CAF adjacent to BAC be bifected by the ftraight line DAE meeting the circumference again in D, and BC the bafe of the fegment produced in E; the excels of BA, AC has a given ratio to AD; and the rectangle which is contained by the lame excefs and the ftraight line ED, is given.

Join BD), and thro' B draw BG parallel to DE meeting AC produced in G. and becaufe BC cuts off from the circle ABC given in magnitude the fegment BAC contain

ing a given angle, BC is therefore gi

D

C

G

ven in magnitude. by the fame rea-
fon BD is given, becaufe the angle
BAD is equal to the given angle EAF;
therefore the ratio of BC to BD is gi- B
ven. and becaufe the angle CAE is e-
equal to EAF, of which CAE is equal
to the alternate angle AGB, and EAF to the interior and oppo-
fite angle ABG; therefore the angle AGB is equal to ABG, and
the ftraight line AB equal to AG; fo that GC is the exccfs of
BA, AC. and because the angle BGC is equal to GAE, that
is to EAF, or the angle BAD, and that the angle BCG is equal
to the oppofite interior angle BDA of the quadrilateral BCAD
in the circle; therefore the triangle BGC is equiangular to EDA

F f

P.

a. 91. Daf

95.

therefore as CC to CB, fo is AD to DB, and, by permutation, 28
GC, which is the excefs of BA, AC to AD, fo is CB to ED.
and the ratio of CB to BD is given;
therefore the ratio of the excefs of BA,
AC to AD is given.

And because the angle GBC is e-
qual to the alternate angle DEB, and
the angle BCG equal to BDE; the tri- B
angle BCG is equiangular to BDE.
therefore as GC to CB, fo is BD to

D

DE, and confequently the rectangle GC, DE is equal to the rec tangle CB, BD which is given, because its fides CB, BD are given. therefore the rectangle contained by the exceis of BA, AC and the ftraight line DE is given.

PRO P. XCIX.

F from a given point in the diameter of a circle given in position, or in the diameter produced, a ftraight line be drawn to any point in the circumference, and from that point a straight line be drawn at right angles to the firft, and from the point in which this meets the circumference again, a ftraight line be drawn paralleł to the first; the point in which this parallel meets the diameter is given; and the rectangle contained by the two parallels is given.

In BC the diameter of the circle ABC given in pofition, or in BC produced, let the given point D be taken, and from D let a ftraight line DA be drawn to any point A in the circumference, and let AE be drawn at right angles to DA, and from the point E where it meets the circumference again let EF be drawn parallel to DA meeting EC in F; the point F is given, as alfo the rectangle AD, EF.

Produce EF to the circumference in G, and join AG. because Cor. 5.4. GEA is a right angle, the ftraight line AG is the diameter of the circle ABC; and BC is alfo a diameter of it; therefore the point H where they meet is the center of the circle, and confe quently H is given. and the point D is given, wherefore DH is given