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lines DB, BC are each of them given; and the ratio of DB to BA is given; therefore AB, BC are given.

The Composition 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 square of one of the sides of the parallelogram, together with the space which has a given ratio to the square of the other side is to be made equal; and let this given ratio be the same which the square of the given straight line MG has to the square of GF.

By the 88th dat. find two straight lines DB, BC, which contain a rectangle equal to the given rectangle MG, GK, and such that the sum of their

squares

is equal to the given rectangle KG, GL; therefore, by the determination of the problem in that proposition, twice the rectangle MG, GK must not be greater than the rectangle KG, GL. Let it be so straight lines DB, BC, in the angle DBC equal to the given angle FGH; and, as MĞ to GF, so make DB to BA, and complete the parallelogram AC; AC is equal to the rectangle FH, GK: and the square

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GH
BE C

K of BC together with the square of BD, which, by the construction, has to the square of BA the given ratio which the square of MG has to the square of GF, is equal, by the construction, to the given rectangle KG, GL. Draw AE perpendicular to BC.

Because, as DB to BA, so is MG to GF; and as BA to AE, so GF to FH; ex æquali, as DB to AE, so is MG to FH; therefore as the rectangle DB, BC, to AE, BC, so 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.

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If a straight line drawn within a circle given in magni

tude cut off a segment which contains a given angle, the straight line is given in magnitude.

In the circle ABC given in magnitude let the straight line AC be drawn, cutting off the segment AEC which contains the given angle AEC; the straight line AC is given in magnitude.

Take D the centre of the circle *, join AD, and pro- * 1. 3. duce it to E, and join EC: the angle ACE being a right * angle, is B given; and the angle AEC is given;

E therefore * the triangle ACE is

D giveu in species, and the ratio of EA to AC is therefore given, and EA

A is given in magnitude, because the circle is given in magnitude : AC is therefore given * in magnitude.

31. 3.

* 43 Dat.

5 Def.

2 Dat.

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If a straight line given in magnitude be drawn within a

circle giden in magnitude, it shall cut off a segment containing a given angle.

Let the straight line AC given in magnitude be drawn within the circle ABC given in magnitude; it shall cut off a segment containing a given angle.

Take D the centre of the circle, B join AD and produce it to E, and

E join EC: and because each of the straight lines EA and AC is given, their ratio is given *: and the angle A ACE is a right angle, therefore the triangle ACE is given * in species, and consequently the angle AEC is given.

* 1 Dat.

* 46 Dat. * 6 Def.

PROP. XCIII.

If from any point in the circumference of a circle given in

position two straight lines be drawn, meeting the circum-
ference 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 also
given.
From any point A in the circumference of a circle
ABC given in position, let AB,
AC, be drawn to the circumfe-

A
rence making the given angle BAC:
if the point B be given, the point
C is also given.

D
Take D the centre of the circle,

BV

C and join BD, DC, and because

each of the points B, D, is given, 29 Dat.,

BD is given * in position; and be-
cause the angle BAC is given, the angle BDC is given *,
therefore because the straight line DC is drawn to the
given point D, in the straight line BD given in position,
in the given angle BDC, DC is given * in position.
And the circumference ABC is given in position, there-
fore * the point C is given.

* 20.3.

* 32 Dat.

* 28 Dat.

91.

PROP. XCIV.

# 29 Dat.

If from a given point a straight line be drawn touching a circle given in position; the straight line is given in position and magnitude.

Let the straight 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 centre of the circle, and join DA, DB: Because each of the points D, A is given, the straight line AD is given * in position and magnitude: and BDA is

right *angle, wherefore DA is С. * Cor. 5. 4. a diameter * of the circle DBA

described about the triangle
DBA; and that circle is there-
fore given in position: and the
circle

BC is given * in position, * 28 Dat. therefore the point B is given *.

18. 3.

a

А

The point A is also given: therefore the straight line
AB is given * in position and magnitude.

** 29 Dat.

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If a straight line be drawn from a given point without

a circle given in position; the rectangle contained by the segments betwixt the point and the circumference of the circle is given.

Let the straight line ABC be drawn from the given point A without the circle BCD given in position, cutting it in B, C; the rectangle BA, AC, is given.

From the point A draw* AD touching the circle; wherefore AD is given * in position and

В А magnitude: and because AD is given, the square of AD is given *, which is equal* to the rectangle BA, AC: therefore the rectangle BA, AC, is given.

17. 3.

* 94 Dat.

* 56 Dat. * 36. 3.

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If a straight line be drawn through a given point within

a circle given in position, the rectangle contained by the segments betwixt the point and the circumference of the circle is given.

Let the straight line BAC be drawn through the given point A, within the circle BCE, given in position; the rectangle BA, AC, is given.

Take D, the centre of the circle, join AD, and produce it to the points E, F: because the points

D A, D are given, the straight line AD is given * in position; and B A C the circle BEC is given in position; F therefore the points E, F, are given *; and the point A is given, therefore EA, AF, . 28 Dat. are each of them given *, and the rectangle EA, AF, * 29 Dat. is therefore given; and it is equal to the rectangle * 35. 3. BA, AC; which consequently is given.

* 29 Dat.

94.

PROP. XCVII.

If a straight line be drawn within a circle given in mag

nitude, cutting of a segment containing a given angle; if the angle in the segment be bisected by a straight line produced till it meets the circumference, the straight lines which contain the given angle shall both of them together have a given ratio to the straight line which bisects the angle. And the rectangle contained by both these lines together which contain the given angle, and the part of the bisecting line cut off below the base of the segment, shall be given. Let the straight line BC be drawn within the circle ABC, given in magnitude, cutting off a segment containing the given angle BAC, and let the angle BAC be bisected 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 straight line ED cut off from AB below BC the base of the segment, is given.

Join BD; and because BC is drawn within the cir

cle ABC given in magnitude cutting off the segment * 91 Dat. BAC, containing the given angle BĂC; BC is given

in magnitude: by the same reason BD is given; there-
fore * the ratio of BC to BD is given : and because the
angle BAC is bisected by AD, as BA to AC, so is *
BE to EC; and, by permutation, as AB to BE, so is
AC to CE; wherefore *, as BA and AC together to
BC, so is AC to CE; and because the angle BAE is
equal to EAC, and the angle
ACE to ADB*, the triangle
ACE is equiangular to the tri-
angle ADB; therefore as AC
to CE, so is AD to DB: but
as AC to CE, so is BA toge-

E
ther with AC to BC; as there-

B fore BA and AC to BC, so is AD to DB; and, by permutation, 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.

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

Because the triangle BDE is equiangular to the tri

1 Dat.

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3. 6.

the 12. 5.

* 21. 3.

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