IF two straight lines contain a given parallelogram in a given angle, and if the square of one of them, together with the space which has a given ratio to the square of the other, be given, each of the straight lines shall be given. Let the two straight lines AB, BC contain the given parallelogram AC in the given angle ABC, and let the square of BC together with the space which has a given ratio to the square of AB be given, AB, BC, are each of them given. Let the square of BD be the space which has the given ratio to the square of AB: therefore, by the hypothesis, the square of BC together with the square of BD is given. From the point A, draw AE perpendicular to BC; and because the angles ABE, BEA, are given, the triangle ABE 43 Dat. is given a in species: therefore the ratio of BA to AE is given: And because the ratio of the square of BD to the square of BA is given, the ratio of the straight line BD to 58 Dat. BA is given; and the ratio of BA to AE is given; there9 Dat. fore the ratio of AE to BD is given, as also the ratio of the rectangle AE, BC, that is, of the parallelogram AC, to the rectangle DB, BC; and AC is given, therefore the rectan gle DB, BC is given; and the square of BC together with • 38 Dat. the square of BD is given: therefored because the rectan gle contained by the two straight lines DB, BC, is given, and the sum of their squares 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 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, and join the straight lines DB, BC in the angle DBC equal to the given angle FGH; and, as MG to GF, so make DB to BA, and complete the parallelogram AC: AC is equal to the rectangle FH, GK; and the square 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 ÃE, BC, that is, the parallelogram AC, is equal to the rectangle FH, GK. PROP. XCI. If a straight line drawn within a circle given in magnitude cuts 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. 88. Take D the centre of the circle, join AD, and produce 1. 3. it to E, and join EC: The angle ACE 31. 3. being a right angle, is given; and the 48 Dat. angle AEC is given; therefore the triangle ACE is given in species, and the ratio of EA to AC is therefore given, A and EA is given in magnitude, because 5 Def. the circle is given in magnitude: AC • 2 Dat. is therefore given in magnitude. B E D Ir a straight line given in magnitude be drawn within a circle given 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, 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 * 1 Dat. givena; and the angle ACE is a right angle, therefore the triangle ACE is A 46 Dat. given in species, and consequently the angle AEC is given. D E IF from any point in the circumference of a circle given in position 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 also given. From any point A in the circumference of a circle ABC given in position, let AB, AC, be drawn to the circumference making the given Take D the centre of the circle, and B * 29 Dat. points B, D, is given, BD is givena in A position; and because the angle BAC is given, the angle 20.3. BDC is given, therefore because the straight line DC is C drawn to the given point D, in the straight line BD given in position in the given angle BDC, DC is given in posi- 32 Dat. tion: And the circumference ABC is given in position, therefored the point C is given. 28 Dat. PROP. XCIV. 1 Ir 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 a in position 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 position: And the circle BC is given in position, there 91. 29 Dat. b 18. 3. A Cór. 5. 4. a 6 Def. fore the point B is given. The point A is also given: 28 Dat. Therefore the straight line AB is given in position and magnitude. PROP. XCV. 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 C is given in position and magnitude: And because AD is given, the square of AD is given, which is 92. D equal to the rectangle BA, AC: Therefore the rectangle à 36. 3. BA, AC, is given. 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, B D E 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 * 29 Dat. straight line AD is given in position; and the circle BEC is given in position; 28 Dat. therefore the points E, F, are given; and the point A is given, therefore EA, AF, are each of them given, and the rectangle EA, AF, 35.3. is therefore given; and it is equal to the rectangle BA, AC; which consequently is given.. 94. PROP. XCVII. F Ir a straight line be drawn within a circle given in magnitude, cutting off 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 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 circle ABC |
