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Fig. 46.

54. On a given right-line to describe a segment of a circle that shall contain a given angle.-Let CD (fig. 46) be the given line, making with CT the angle DCT equal to the given angle. Draw COG perpendicular to CT, and draw DO, making with DC an angle ODC equal to the angle OCD. From O as a centre, and with OC or OD as radius, describe a circle cutting COG in the point G;

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join G D. Then the angle CGD is equal to the given. angle DCT.

If we take any other points as F, H, on the circumference and join them with C and D, the angles CFD, CHD will be each equal to the angle CGD, or to the given angle DCT. The angle COD at the centre of the circle is double the angle CGD, and double the angle DCT.

If AB is the given line, and the given angle is a right angle, bisect A B in O, and from O as a centre, and with OA as radius, describe a circle. Take any point F on the circle and draw the chords FA, FB; then the angle AFB is a right angle.

55. From the extremities of a given right-line to draw two other lines in any directions meeting at right angles to each other.-Let AB (fig. 46) be the given line, and let BF be the given length of one of the other lines less than A B. Bisect A B in O, and from O as a centre describe a semicircle on AB. From B as a centre, and with BF as radius, draw an arc cutting the circle in F; join AF: then A FB is a right angle.

56. To lay out an arc of a circle as a railway curve of given radius, by finding points on the circumference.-Let

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AT (fig. 47) be the tangent to the proposed curve at the point A, O the centre of the circle. Suppose it is required to find points on the arc at distances apart equal to a given length AT. From A as a centre, and with AT as radius, draw an arc TB, and construct the angle TAB equal to

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half the angle AOB at the centre of the circle. Then B is the first point on the curve. Make the angles BAC, CAD, DAE, each equal to BAT; and from B as a centre, with BA as radius, draw an arc cutting AC in C; from C as a centre, and with CB as radius, draw an arc cutting A D in D; from D as a centre, and with DC as radius, draw an arc cutting AE in E; then A, B, C, D, and E, are all upon the required curve.

This method is based on the principle that angles which stand upon equal arcs of a circle and have their vertices on the circumference are all equal.

57. To find arithmetically the radius of an arc of a circle of which the chord and height are given.-Let BC · (fig. 45) be the given chord whose length is known arithmetically, AG the given height. Bisect BC in G. Add together the squares of the lengths A G and B G, and divide the sum by twice AG, the quotient is the radius of the circle. For example, let BC equal 10, AG equal 2;

BG,

5×5+2×2=29, 2 × 24, 29 divided by 4 is 74 the length of the radius.

58. To describe a circle touching a given right-line and passing through two given points. First, let the line joining the two points be not parallel to the given line. Let EC (fig. 48) be the given line, P and Q the given points. Join Fig. 48.

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P and Q, and produce PQ to meet the given line in C. Produce PC to G, making GC equal to CQ. On PG describe a semicircle, and draw CF perpendicular to PC, meeting the circle in F. Take CE equal to CF, and draw EX perpendicular to CE. Bisect PQ in D, and draw DX perpendicular to PQ, and

meet

Then

ing EX in the point X.
the circle described from X as a
centre, and with XE for radius,
will touch the given line at E and
pass through the points P and Q.

Secondly, let the line joining the points be parallel to the given line. Let EC (fig. 49) be the

Fig. 49.

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given line, P and Q the given points. Join PQ, and bisect

PQ in D. Draw DE perpendicular to PQ, and join EP. Bisect EP in F, and draw FX perpendicular to EP, meeting DE in X. Then the circle described from X as a centre, and with XE for radius, will touch the given line at E and pass through the given points P and Q.

59. To describe a circle touching a given circle, and also a given right-line outside of it in a given point.-Let O (fig. 50) be the centre of the given circle, A B the given

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line wholly outside of the circle, A the given point therein. Draw AF perpendicular to the line AB, and also a diameter COD parallel to AF. Join DA and CA, producing AC to meet the circle again at E. Join EO, and produce EO to meet AF in G; join ED, producing it to meet A F in F. Then G F is equal to GA or to GE; and from G as a centre, with G A as radius, draw a circle which will touch the given circle in E, and the given line at the given point A.

Again, let A D cut the given circle at H; join OH, and

produce OH to meet AF in I. Then IH is equal to I A; from I as a centre, and with IA as radius, describe a circle which will touch the given circle at H and the given line at A.

60. To describe a circle of given radius which shall touch two given circles. First, suppose the given circles do not intersect, their distance apart being not greater than the diameter of the required circle.

Let A, B (fig. 51) be the centres of the given circles,

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joined by the indefinite right line D F, which cuts the circumferences in C, M, N, and E. Take CD, EF, each equal to the given radius; from A as a centre, and with AD as radius, describe a circle; and from B as a centre, with BF as radius, draw arcs cutting the last circle in P and Q. Join the points A, P, B, Q; then a circle drawn from either P or Q as centre, and with the given length of radius, will touch the given circles in G and H or in K and I.

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