as radius, describe a circle cutting CD in O. Draw the line APOQ meeting CF in P and BD in Q. circle described from O as a centre, and with OP or OQ as radius, will touch all four of the given arcs. 63. To describe a circle which shall touch a given circle and a given right-line, and shall also have its centre on another given line.-Let O (fig. 60) be the centre and OD the radius of the given circle; A B the given line to be touched, P Q the given line on which the centre of the circle is to be found. Let PQ cut A B in the point A. From O draw ODB perpendicular to AB, and produce it to C, making BC equal to OD. Draw CP parallel to AB, meeting P Q in P; join PO, and produce PO indefinitely to R. Take any point Q on the line PQ, and drop therefrom the line QF perpendicular to CP; from Q as a centre, and with Q F as radius, describe a circle cutting PR in R, and join Q R. Draw OHX parallel to R Q, meeting PQ in X, and draw X G perpendicular to A B. Then a circle described from X as a centre, and with X G or X H as radius, will touch A B at G and the given circle at H, as required. 64. To describe a circle touching a given circle, and also two given right-lines which are not parallel.-Let A K and A B (fig. 60) be the given lines meeting at the point A; O the centre and OD the radius of the given circle. Bisect the angle BAK by the line PA Q, then the centre of the required circle must lie upon this line. Find X the centre and X G the radius of the circle which touches the given circle and also the line A B at G (63); draw X K perpendicular to A K, and the circle drawn from X as a centre, with XG or XK or X H as radius, touches the two lines in G and K and the given circle in H. INSCRIBED AND CIRCUMSCRIBED CIRCLES. 65. To inscribe three circles in an equilateral Gothic arch.-Let A DB (fig. 61) be the arch, having the arcs AD, BD struck from B and A as centres. Draw the horizontal springing line AB; it is required to inscribe three circles in the figure A B D. Bisect Draw CX to meet GE in F, From H as a centre, and arc cutting B F in I, and Bisect A B at C and draw the perpendicular C D. the angles ACD, BCD by the lines C X, CY (8). A G, BE perpendicular and equal to AB; and draw EGF parallel to A B. Produce and cutting A G in H; join B F. with HG as radius, describe an join HI. Draw BP parallel to HI and cutting C F in X; take CY equal to CX and draw X Y, cutting CD in Q. Draw XL perpendicular to A B, and the circle drawn from X as a centre, with X P or XL or XQ as radius, will touch AB at L, CD at Q, and AD at P. Also a circle described from Y as a centre, and with the same radius, will touch CD at Q, and also the arc BD and the line BA. We have now to draw the circle which shall touch the circles X and Y and also the arcs AD, BD. Produce BP (fig. 62) to R making PR equal to PX; and from B as a centre, with BR as radius, Fig. 62. O as a centre, and with OE or OF as radius, will touch the arcs AD, BD, and also the two previously inscribed circles. 66. To inscribe three circles in an equilateral triangle.— Let ABC (fig. 63) be the triangle. Bisect each side at scribe a circle cutting X A in P, XC in Q. Draw the lines OP, OQ, PQ; let OQ intersect A F in R, and the circles described from O, P, and Q as centres, with OR, or half OQ, for radius, will be inscribed in the triangle. 67. To inscribe three circles in a given isosceles triangle. -In this case two only of the circles will be equal. Let ACB (fig. 64) be the given Fig. 64. triangle. Bisect the base A B at D and draw DC perpendicular thereto. Bisect each angle CAB, CDA, CDB, CBA, by the lines A X, DX, BY, DY. Draw XY cutting DC in M; then the circles drawn from X and Y as centres, and with XM or Y M for radius, will touch each other and the sides of the triangle. The centre of the third circle will lie in the line DC. Draw X N perpendicular to AC and produce it to T, making NT equal to NX. Draw TH parallel to A C. Find on the line DC (58) the centre Z of the circle which touches TH at H, and also passes through the points X, Y. Draw Z H perpendicular to A C and cutting it at E. Join XZ, YZ; and the circle drawn from Z as a centre, with ZE or ZF or Z K as radius, touches the sides A C, BC of the given triangle, and also the two previously inscribed circles at the points F and K. 68. In a given triangle to inscribe a circle.-Let A B C (fig. 65) be the given triangle. Bisect the angles CAB, CBA by the lines AO, BO, meeting in the point O. Draw O D perpendicular to A B, and the circle drawn from O as a centre, with OD for radius, will touch each side of the triangle. 69. In an equilateral triangle to inscribe a circle.-Let ABC (fig. 66) be the given triangle. Bisect any two sides AC, BC by the points D and E. Draw BD, A E, intersecting at O. Then the circle described from O as a centre, and with OD or O E as radius, will touch the sides of the E |
