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Let A and B be the given points, and CD the given line. Join BA, and produce it to meet CD in D. Take DC a mean proportional between DA and DB. Cis the point required.

Join AC, BC; and about the triangle ABC describe a circle; DC is a tangent at the point C (Eucl. iii. 37.), and.. the angle is the greatest (ii. 62.).

(5.) To determine the position of a point, at which lines drawn from three given points, shall make with each other angles equal to given angles.

A

D

B

E

Let A, B, C be the three given points; join AB, and on it describe a segment of a circle containing an angle equal to that which the lines from A and B are to include. Complete the circle, and make the angle ABD equal to that which the lines from A and C are to include. Join DC, and produce it to the circumference in E. E is the point required.

Join AE, BE. Then the angle AEC=ABD, and AEB is of the given magnitude, by construction.

(6.) To divide a straight line into two parts such, that the rectangle contained by them may be equal to the square of their difference.

Let AB be the given line; upon it describe a semicircle ADB. From B draw BC at right angles and

equal to AB. Take O the centre, and join OC; and from D draw DE perpen'dicular to AB; AB is divided in the point E, as was required.

Since BC is double of BO, DE is double of OE (Eucl. vi. 2.), and OE

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being half the difference between AE and EB, DE is equal to the difference. Also (Eucl. vi. 13.) the rectangle AE, EB is equal to the square of DE.

(7.) If a straight line be divided into any two parts; to produce it, so that the rectangle contained by the whole line so produced, and the part produced may be equal to the rectangle contained by the given line and one segment.

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D

E

BG

Let AB be the given line divided into two parts in the point C. On AB as diameter describe a circle ADB. From B draw BE at right angles to AB, and .. a tangent to the circle; and make BE a mean proportional between AB and AC. Take O the centre; join EO, and produce it to F. Produce AB to G, making BG equal to ED. Then will the rectangle AG, GB be equal to the rectangle BA, AC.

Since DE = BG, the rectangle BG, GA is equal to the rectangle DE, EF, i. e. to the square of EB, or to the rectangle AB, AC, by construction.

COR. 1. If it be required to produce the line, so that the rectangle contained by the whole line produced and the part produced, may be equal to the rectangle con

tained by two given lines; find BE a mean proportional between the two given lines, and proceed as in the proposition.

COR. 2. If it be required to produce the line, so that the rectangle contained by the whole line produced and the part produced, may be equal to a given square; take BE equal to a side of the square, and proceed as in the proposition.

(8.) To determine two lines such that the sum of their squares may be equal to a given square, and their rectangle equal to a given rectangle.

Let AB be equal to a side of the given square. Upon it describe a semicircle ADB; and from B draw BC perpendi

A

cular to AB, and equal to a fourth proportional to AB and the sides of the given rectangle. From C draw CD parallel to BA. Join AD, DB; they are the lines required.

Since CB touches the circle at B, the angle CBD is equal to DAB, and the angles DCB, ADB are right angles; .. the triangles DCB, ADB are equiangular, and AB AD :: DB : BC,

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whence the rectangle AD, DB is equal to the rectangle AB, BC, i. e. to the given rectangle. Also the squares of AD, DB are equal to the square of AB, i. e. to the given square.

(9.) To divide a straight line into two parts, so that

the rectangle contained by the whole and one of the parts may be equal to the square of a given line, which is less than the line to be divided,

Let AB be the given line to be divided. Upon it describe a semicircle, in which place the line AC to the given line.

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Join CB; and on it describe a semicircle
CDB, cutting AB in D; D is the point required.

B

Since the angle ACB is in a semicircle, it is a right angle, .. AC touches the circle CDB (Eucl. iii. 16. Cor.); whence the rectangle BA, AD is equal to the square of AC, i. e. to the square of the given line.

(10.) To divide a given line into two such parts that the rectangle contained by the whole line and one of the parts may be (m) times the square of the other part, m being whole or fractional.

B

E

Let AB be the given line, and in it produced, take BC=an mth part of AB. On AC describe a semicircle, and from B draw BD perpendicular to AC. Bisect CB in O; join OD, and take OE=OD; and AB will be divided in E, as required.

On BC describe a semicircle, cutting OD in F; join FE. Then the angle DOE being common to the triangles DOB, EOF, and DO, OB respectively equal to EO, OF, the triangles will be similar and equal, and .. the angle OFE equal to OBD, and .. a right angle; whence FE is a tangent to the circle CFB. Hence the rectangle AB, BC is equal to the square of DB, i. e. to the square of FE, or the rectangle CE, EB.

From each of these equals take away the rectangle CB, BE; and the rectangle AE, CB is equal to the square of BE, .. (m) times the rectangle AE, CB, i. e. the rectangle AB, AE is equal to (m) times the square of BE.

(11.) To divide a given line into two such parts that the square of the one shall be equal to the rectangle contained by the other and a given line.

Let AB be the given line to be divided, (see last Fig.) and BC the other given line. Let them be placed so as to be in the same straight line. On AC describe a semicircle and draw the lines, as in the last proposition; and E is the point required.

For the rectangle AE, CB is equal to the square BE

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(12.) A straight line being given in magnitude and position; to draw to it from a given point, two lines, whose rectangle shall be equal to a given rectangle, and which shall cut off equal segments from the given line.

D E/G

Let AB be the given line, and C the given point. Bisect AB in D, and from D draw DO at right angles to AB, and let fall the perpendicular CE. With the centre C, and radius equal to a fourth proportional to 2 CE and the sides of the given rectangle, describe a circle cutting DO in O. Join OC; and with the centre 0,

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