IF two ftraight lines contain a given parallelogram in a given angle, and if the fquare of one of them together with the fpace which has a given ratio to the fquare of the other be given, each of the straight lines fhall be given.

Let the two ftraight lines AB, BC contain the given parallelogram AC in the given angle ABC, and let the fquare of BC together with the fpace which has a given ratio to the square of AB be given, AB, BC are each of them given.

Let the fquare of BD be the space which has the given ratio to the fquare of AB; therefore, by the hypothefis, the fquare of BC together with the fquare of BD is given. From the point A, draw AE perpendicular to BC; and because the angles a 43. dat. ABE, BEA are given, the triangle ABE is given in fpecies; therefore the ratio of BA to AE is given: And because the ratio of the fquare of BD to the fquare of BA is given, the ra58. dat. tio of the ftraight line BD to BA is given ; and the ratio of BA to AE is given; therefore the ratio of AE to BD is given, as alfo the ratio of the rectangle AE, BC, that is, of the parallelogram AC to the rectangle DB, BC; and AC is given, therefore the rectangle DB, BC is given; and the square of

c 9. dat.

& 88. dat. BC together with the fquare of BD is given; therefore & because the rectangle contained by the two ftraight lines DB, BC is given, and the fum of their fquares is given: The straight lines DB, BC are each of them given; and the ratio of DB to BA is given; therefore AB, BC are given.

The compofition is as follows.

Let FGH be the given angle to which the angle of the parallelogram is to be made equal, and from any point F in GF draw FH perpendicular to GH; and let the rectangle FH, GK be that to which the parallelogram is to be made equal; and let the rectangle KG, GL be the space to which the fquare

of

of one of the fides of the parallelogram together with the space which has a given ratio to the fquare of the other fide, is to be made equal; and let this given ratio be the fame which the fquare of the given ftraight line MG has to the fquare of GF.

By the 88th dat. find two ftraight lines DB, BC which contain a rectangle equal to the given rectangle MG, GK, and fuch that the fum of their fquares is equal to the given rectangle KG, GL; therefore, by the determination of the problem in that propofition, twice the rectangle MG, GK muft not be greater than the rectangle KG, GL. Let it be fo, and join the ftraight lines DB, BC in the angle DBC equal to the given angle FGH; and, as MG to GF, fo make DB to BA, and complete the parallelogram AC: AC is equal to the rect

angle FH, GK; and the fquare of BC together with the fquare of BD, which, by the construction, has to the fquare of BA the given ratio which the fquare of MG has to the fquare of GF, is equal, by the conftruction, to the given rectangle KG, GL. Draw AE perpendicular to BC.

Because, as DB to BA, fo is MG to GF; and as BA to AE, fo GF to FH; ex aequali, as DB to AE, fo is MG to FH; therefore, as the rectangle DB, BC to AE, BC, fo is the rectangle MG, GK to FH, GK; and the rectangle DB, BC is equal to the rectangle MG, GK; therefore the rectangle AE, BC, that is, the parallelogram AC, is equal to the rectangle FH, GK.

PROP. XCI.

IF a ftraight line drawn within a circle given in magnitude cuts off a fegment which contains a given angle; the straight line is given in magnitude.

88.

In the circle ABC given in magnitude, let the ftraight line AC be drawn, cutting off the fegment AEC which contains the given angle AEC; the ftraight line AC is given in magnitude. Take D the centre of the circle, join AD and produce it a 1. 3.

b 31. 3.

to E, and join EC: The angle ACE being a right angle is given; and the angle c43. dat. AEC is given; therefore the triangle ACE is given in fpecies, and the ratio of EA to AC is therefore given; and EA is d5. def. given d in magnitude, because the circle is given in magnitude; AC is therefore given in magnitude.

e 2. dat.

89,

a I. dat.

IF

PROP. XCII.

B

D

C

a ftraight line given in magnitude be drawn within a circle given in magnitude, it fhall cut off a fegment containing a given angle,

Let the ftraight line AC given in magnitude be drawn within the circle ABC given in magnitude; it fhall cut off a fegment containing a given angle.

Take D the centre of the circle, join AD and produce it to E, and join EC: And because each of the straight lines EA, and AC is given, their ratio is given; and the angle ACE is a right angle, therefore b 46. dat, the triangle ACE is given in fpecies, A and confequently, the angle AEC is given.

B

E

from any point in the circumference of a circle gi

Iven in polition two straight lines be drawn meeting

the circumference and containing a given angle; if the point in which one of them meets the circumference again be given, the point in which the other meets it is alfo gi

ven.

From any point A in the circumference of a circle ABC given in pofition, let AB, AC be drawn to the circumference making the given angle BAC; if the point B be given, the point C is alfo given.

a

Take D the centre of the circle, and join BD, DC; and because each of the a 29. dat, points B, D is given, BD is given in pofition; and because the angle BAC is given, the angle BDC is given, therefore

b.20 3.

because the straight line DC is drawn to the given point D in the ftraight line BD given in pofition in the given angle BDC, DC is given in pofition: And the circumference ABC is gi-c 32. dat. ven in pofition, therefore the point C is given.

1F

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

B

Take D the centre of the circle, and join DA, DB: Because each of the points D, A is given, the ftraight line AD is given a in pofition and magnitude: And DBA is a right angle, wherefore DA is a diameter

of

the circle DBA, described about the triangle DBA; and that circle is therefore given in pofition: And the circle BC is given in pofition, therefore the point B.

d 28. dat.

91.

a 29 dat. b 18. 3. c Cor. 5. 4.

is given: The point A is alfo given; therefore the straight e 28. dat. line AB is given in pofition and magnitude.

IF

PRO P. XCV.

Fa ftraight line be drawn from a given point without a circle given in pofition; the rectangle contained by the segments 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 BCD given in pofi

tion, cutting it in B, C; the rectangle

BA, AC is given.

а

From the point A, draw AD touching the circle; therefore AD is given bin pofition and magnitude: And becaufe AD is given, the fquare of AD is

с

d

92.

a 17. 3.

BA

b 94. dat.

given which is equal to the rectangle BA, AC: Therefore e 56. day the rectangle BA, AC is given.

93.

a 19. dat.

b 28. dat.

C 35. 3.

94.

PROP. XCVI.

IF a ftraight line be drawn through 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 through the given point A within the circle BCE given in pofition; the rectangle BA, AC is given.

D

Take D the centre of the circle, join AD and produce it to the points E, F. Because the points A, D are given, the ftraight line AD is given in pofition; and the circle BEC is given in pofition; therefore the points E, F are given, and B the point A is given, therefore EA, AF are each of them given ; and the rectangle EA, AF is therefore given; and it is equal to the rectangle BA, AC, which confequently is given.

PROP. XCVII.

F

A

C

Fa 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 straight 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 these lines together which contain the given angle, and the part of the bifecting line cut off below the bafe of the fegment, fhall be given.

Let the flraight 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 straight 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.

Join BD; and becaufe BC is drawn within the circle ABC