CD equal to the given radius. Through D draw DO parallel to CB; O is the centre of the circle required.

Through O draw OE parallel to DC; .. CO is a parallelogram; whence OE is equal to DC, i. e. to the given radius. With the centre O, and radius OE, describe a circle; it will touch CB in E, because CO being a parallelogram, and ECD a right angle, CEO is also a right angle.

(40.) To describe a circle, which shall touch a straight line in a given point, and also touch a given circle.

Let AB be the given line, and C the given point in it, O the centre of the given circle. Draw CD perpendicular to AB, and OE parallel to CD. Join CE, meeting the circumference in F. Join OF, and produce it to meet CD in D. D is the centre of the circle required.

E

Since the triangles OEF, CFD are similar, and OE =OF, .. FD=DC; consequently a circle described with the centre D, and radius DF, will pass through C, and touch AB in C, because the angles at C are right angles; and it will touch the given circle in F, since the line joining the centres passes through F.

B

(41.) To describe two circles, each having a given radius, which shall touch each other, and the same given straight line on the same side of it.

Let AB be the given straight line. From any point A in it, draw AC at right angles to it, and make AC, AD, equal to the given radii. Produce CA to E, making AE=AD. Draw DO parallel to AB; and with the centre C,

and radius CE, describe a circle cutting DO in 0. C and O will be the centres of the circles required.

Join CO; and draw OB perpendicular to AB; then DAB being a right angle, as also ABO, .. AD is parallel to BO; and DO was drawn parallel to AB, :. AO is a parallelogram, and OB= AD. With the centres C and O, and radii CA, OB describe circles, they will touch AB, since the angles at A and B are right angles; they will also touch each other, for CO is equal to CE, or to CA and AE, i. e. to CA and AD, or the sum of the radii.

(42.) To describe a circle passing through two given points, and touching a given circle.

Describe a

Let A and B be the given points, and CDE the given circle. circle through A and B, and cutting the given circle in D and E. Join DE, EB, DA, AB. Then the angle EDA =EBA; if . DE and BA be pro

duced to meet in F, the triangle FDA will be similar to the triangle FBE;

and.. DF: FA :: BF : FE,

or the rectangle DF, FE is equal to the rectangle AF, FB. Draw FG a tangent to the given circle; then the

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

E

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.

DD

(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

B

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

proportional between DE and DḤ; 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.

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 circle, whose centre is 0. Draw HI, KL parallel to the given lines, so that their perpendicular distances from those lines may be equal to OF the radius of the given circle. By

E

the last problem describe a circle touching HI, KL,