PROBLEM xx. xxiv.] MATHEMA TICS PR ACTI C AL GEOMETRY. 605 



other chord A E, making the angle DAE equal to the 



angle BAD; with the centre D and radius D E de 



Fig. 37. 



scribe an arc cutting A B in F ; then F is the point in 

 A B, through which the arc A E C, if continued, would 

 pass. 



When the centre it either not kiwwn, or at too great a 

 distance to be conveniently used (as very frequently occurs 

 in practice), this problem will be found exceedingly useful. 



PROBLEM XX. 



To draw a tangent to a given circle (ABC) through a given 

 point D in the circumference. (Euclid, Book III., 

 Prop. XVII.) 



IST METHOD. If the centre O of the given circle 

 ABC (Fig. 38) be not given, find it by Problem XVII. ; 

 join () J), and through the given point D draw E F per- 

 pendicular to O D ; then E F is the tangent required. 



Fig. S8. Fig. 30. 



Fig. 40. 



2ND METHOD. Take any other point O (Fig. 39) in 

 the circumference of the 

 given circle ABC; join 

 I) O and produce it ; with 

 the centre O, and radius 

 ( ) 1 ) describe a circle cut- 

 ting the given circle in C, 

 and DO produced in G; 

 make the arc G E equal to 

 the arc G C ; join ED and 

 produce it ; then E F is 

 the tangent required. 



SBD METHOD. Take any 

 other point O (Fig. 40) in 

 the circumference of the 

 given circle ABC; join O D 

 and produce it ; with the 

 centre D, and radius D O, 

 describe a circle cutting the given circle in A, and O D 

 produced in G ; with the centres A and G, and any 

 equal radii, describe arcs intersecting each other in the 

 point F; join FD and produce it; then E F is the 

 tangent required. 



PROBLEM XXI. 



T ' <lrnw a tangent to a (riven circle (ABC) through a given 

 point (D) without the circumference. (Euclid, Book 

 III., Prop. XVII.) 



Isr METHOD. Find the centre O (Fig. 41) of the given 

 circle ABC; join O 1 >, and bisect it in G ; with the 

 centre G, and radius GO or G 1), describe a semicircle, 

 cutting the given circle A B C in the point C, which is 

 the point of contact ; join D C and produce it ; then E D 



is the tangent required, touching the circle at the point 

 C, and drawn from the point D. 



Fig. 41. Fig. 42. 



' 2ND METHOD. With the given point D (Fig. 42) as a 

 centre, and any radius, describe an arc cutting the given 

 circle A B C in B and C ; join D B, D C', and produce 

 them till they meet the circumference of the given circle 

 A B C in F and G ; join B G and C F, cutting each other 

 in the point H ; join B C, and through H draw a line 

 parallel to B C, and meeting the circle in the points K L ; 

 draw from the point D lines D E and D M, passing 

 through K and L, which will be tangents to the circle 

 AB(J. 



SRD METHOD. From the given point D (Fig. 43) draw 

 any three lines cutting the Fig. 43. 



given circle in E, A in F, B 

 and in G, C respectively ; join 

 G B, F C, intersecting in K, 

 and G A, EC, intersecting in 

 H ; draw through H and K 

 a line meeting the circle in the 

 points L, >I which will be 

 the points of contact ; and 

 lines drawn from D through 

 those points L, M will be 

 tangents to the circle ABC. 



PROBLEM XXII. 



To aetcribe a circle that shall touch two gtren straight lines 

 (A B and C D) nnt in the lame strait/lit line, and touching 

 at a gicen point (E) in one of them '(A B). 



Fig. 44. 



Produce the given lines, AB 

 and C D (Fig. 44), till they 

 meet in the point G ; make 

 G F equal to G E : from E and 

 F draw E O and F O perpen- 

 dicular to A B and C D, re- 

 spectively intersecting in the 

 point O ; and with the centre 

 O, and radius OE or OF, 

 describe the circle HEF, which 

 will touch the given lines A B, 

 C D, and also A B in the point 

 E, as required. 



PROBLEM XXIII. 



To describe a circle that shall touch a given straight line 

 (AB) in a given point (E), and 

 lhall also pass through another 

 given point (F) not in the same 

 straight line (A B). 



From the given point E (Fig. 45) 

 draw E G perpendicular to A B ; 

 join E F, and bisect it by a per- 

 pendicular C D, cutting E G in the 

 point O, with which as a centre, 

 and the radius O E, describe the 

 circle E D F, which will touch the 

 line A B in the given point E, and 

 also pass through the given point F. 



PROBLEM XXIV. 



To describe a part of a circle that shall touch a given 



Fig. 45. 



