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square of FG is equal to the rectangle EF, FD, and .. to the rectangle BF, FA; whence a circle described through the points A, G, B, will touch the given circle, since it touches FG.

(43.) To describe a circle, which shall pass through a given point, and touch a given circle and a given straight line.

Let ABC be the given circle, D the given point, and EF the given straight line. Through O draw AOE perpendicular to EF. Join AD; and divide it in G, so that the rectangle AG, AD, may be equal to the rectangle AC, AE. Through

GH

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G and D describe a circle touching EF in F; this will also touch the circle ABC.

Draw the diameter FH; it is (Eucl. iii. 18.) parallel to AE. Join AF, meeting the circle in B. Join CB. The triangles ABC, AEF having the angle at A common, and the angles ABC, AEF right angles, are similar; whence

AC: AB :: AF : AE,

... the rectangle AB, AF is equal to the rectangle AC, AE, i. e. to the rectangle AG, AD; .. B is a point in the circle HDF. Take I the centre; join OB, BI. Since AC is parallel to FI, the angle OAB=BFI; but OAB OBA, and IFB = IBF, . OBA= IBF; and OBI is a straight line, which joins the centres of the two circles, which .. touch each other.

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D D

(44.) To describe a circle which shall touch a straight line and two circles given in magnitude and position.

Let A and B be the centres of the two circles, and CD the line given in position. From B let fall the perpendicular BE, and produce it, making EF= the radius of the circle whose centre is A. Through F draw FG parallel to CD. With the centre B, and radius equal to

H

B

G

the difference of the radii of the two circles, describe a circle; through A let a circle be described, touching the line GF and the last described circle (vi. 43.); and let G and H be the points of contact. The centre of this circle will also be the centre of the circle required.

Let O be the centre; join OA, OG, OH; and with the centre O, and radius OI, describe the circle IKL. Since LG=KH=AI, .. OL=OK=OI; the circle IKL :. touches CD in L, and the circle, whose centre is A, in I; and since OB is equal to the difference between OH and HB, i. e. between OA and (IA- BK), or is equal to OK and KB together, .. it touches the circle whose centre is B, in K.

(45.) To describe a circle which shall touch two given straight lines, and pass through a given point between them.

Let AB, CD be the given lines, and E the given

point.

Produce the lines to meet in F. Bisect the angle BFD by the line FG; and from E draw EG perpendicular to FG, and produce it both ways to B and D. Take GH-GE; and make DI a mean

A

E

proportional between DE and DH; a circle described through the points H, E, I, will touch CD.

For the rectangle DE, DH, is equal to the square of DI. And for a similar reason it will touch AB; since the rectangle BH, BE, is equal to the rectangle ED, DH.

H

If the lines AB, CD be parallel; through the given point E, draw DEHB perpendicular to AB or CD; bisect it in G, and make GH = GE. Take DI a mean proportional between DE and DH; and a circle described through I, E and H will be the circle required.

(46.) To describe a circle which shall touch two given straight lines, and also touch a given circle.

Let AB, CD be the given straight lines, EFG the given.

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M

H

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the last problem describe a circle touching HI, KL,

and passing through O the centre of the given circle. Let P be the centre of this circle; it will also be the centre of the circle required.

Join PM, PN, PO. Since these lines are equal, and MQ, RN, OF are also equal by construction, .. PQ, PR, PF are also equal; and a circle described from the centre P at the distance of any one of them, will pass through the extremities of the other two, and touch the lines AB, CD, in Q and R; since the angles at those points are equal to the angles at M and N, and .. right angles; and it will also touch the circle EFG in F, since OP the line joining the centres passes through F.

(47.) To describe a circle which shall touch a circle and straight line, both given in position, and have its centre also in a given straight line.

Let the circle whose centre is A, and the straight line BC be given

in position; and let CD be the line, in which the centre of the required circle is to be. On BC let fall the perpendicular AB; and make BF

=

F

B

E

G

AE; through F draw FG parallel to BC, meeting DC in G.

Join GA; and draw CH parallel to it, meeting the given circle in H, (if the problem be possible). Join AH, and let it meet DC in O. O is the centre of the circle required.

Let fall the perpendicular OI. Then (Eucl. vi. 2.)

HO OC: AH: GC: FB: GC by construction,

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.. HO=10; and a circle described with the centre O, and radius OI or OH, will pass through the extremity of the other, and touch the line BC in I, and the circle in H; because the angles at I are right angles; and 40 the line joining the centres of the circles passes through H.

(48.) Through two given points within a given circle, to describe a circle, which shall bisect the circumference of the other.

Let A and B be the given points within the circle whose centre is. 0. Join AO; and produce it indefinitely; and from O draw OC at right angles to it. Join AC; and draw CD at right

angles to it, meeting AQ produced in D; and through A, B, D describe a circle; it will bisect the other in the points E, and F.

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..the rectangle AO, OD is equal to the square of OC, i.e. to the rectangle EO, OF; whence (Eucl. iii. 35.) EOF is a straight line; and since it passes through the centre of the circle ECF, it will be a diameter of that circle; the circumference ECF is equal to the circumference EGF, or the circumference of the given circle is bisected.

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