DEM. From any Point, as F, draw FA and FB. The Vertex C will be the Center, and CA, or CB, the Radius of the Circle required. The Angles being given, under which three Objects, fituate in a Right Line, are feen, and their Distances from each other known; to determine the Point from which they are feen. A, B, and C are the three Objects. Make the Angles ACD, CAD, alternately, equal to the given Angles, i. e. make ACD equal to the Angle under which AB is feen, and CAD equal to the other Angle given. D B by Pr. 40. Produce CD and AD, interfecting at D. Defcribe a Circle through the two extreme Objects, A & C, and the Angle D; Draw DB, and produce it, till it cuts the oppofite Circumference, at E, the Point fought. DEM. For (having drawn AE and CE) the Angle AEB ACD, and the Angle CEB CAD. P. 10. 3. COR. Hence, through three given Points (not lying in a Right Line) as A, B, and D, the Circumference of a Circle may be defcribed. whofe Center (being found, as above) is C. To draw a Tangent to a Circle, through a given Point in the Circumference. And, to find the Point of Contact of a Tangent to a Circle. Firft; B is the Point given; through which a Tangent is required to be drawn. Having found A, the center of the Circle (by the foregoing) join the point B, and the center of the Circle by a Right Line, AB. At the Point B, make a Right Angle, 2nd. CD is the Tangent given; to find the Foint of Contact. Draw a Perpendicular AB to the Tangent, from the cen ter of the Circle; cuting CD in B ; Pr. 7. B is the Point of Contact, in which the Tangent, AD, touches the Circle. C. 3. 8. 3. PRO. PROBLEM XLII. 17. III. Euclid. To draw a Tangent to a given Circle, from a Point given without the Circle; i. e. to determine the Point, in which a Right Line drawn from the given Point fhall touch the Circle. From the given Point, A, draw AC; to the Center of the Circle. Bifect AC; and, on the point of bisecti- DEM. Having drawn AD and CD, the Otherwife. Join the Point A, and the Center C, as before, cuting the Circumference in B. With the Radius CA defcribe the Ark AD. Draw the Perpend. BD, cuting AD at D. Laftly, draw CD, cuting the given Circle at E, the Point fought. Draw AE. 13 P. 12. 3. A The Right Line AE will touch the Circle, at E. C. 2. 8. 3. D DEM. The Triangles AEC, BDC are congruous. B For, the fides AC, CE are equal to DC and CB, refpec- But CBD is a Right Angle (Con.) wh. AEC is a R.Angle. Therefore, AE touches the Circle at E. M C. 2. S. 3. PRO PROBLEM XLIII. 34. III. Euclid. To cut off a Segment of a Circle, which fhall contain an Angle equal to a given one. X DEC is the Circle given, and X the given Angle. E Draw a Tangent, AB, to the given Circle. At the Point of contact, C, make the Angle ACD equal to X, the Angle given; and DEFC is the Segment required. QE F. F DEM. For, if from any Point in the Circumference, as E or F, Lines are drawn to the extremes of the Chord CD; the Angle DEC or DFC, is equal to ACD, (equal X, by Con.) P. 13.3. A B PROBLEM XLIV. 33. III. Euclid. On a given Line, to defcribe a Segment of a Circle, which fhall contain an Angle equal to a given one. AB is the given Line, and X the given Angle. Make an Angle, BAD, equal to the X F DEM. From any Point, as F, draw FA and FB. The Vertex C will be the Center, and CA, or CB, the Radius of the Circle required. The Angles being given, under which three Objects, fituate in a Right Line, are feen, and their Distances from each other known; to determine the Point from which they are feen, A, B, and C are the three Objects. Make the Angles ACD, CAD, alternately, equal to the given Angles, i. e. make ACD equal to the Angle under which AB is feen, and CAD equal to the other Angle given. D B by Pr. 40. Produce CD and AD, interfecting at D. Defcribe a Circle through the two extreme Objects, A & C, and the Angle D; Draw DB, and produce it, till it cuts the oppofite Circumference, at E, the Point fought. DEM. For (having drawn AE and CE) the Angle AEB ACD, and the Angle CEB⇒CAD. M 2 P. 10. 3. PRO |