DC is drawn to the given point D in the ftraight line BD given in e. 32. Dat. position in the given angle BDC, DC is given in pofition. and the d. 28. Dat. circumference ABC is given in pofition, therefore the point C is given. 91. IF PROP. XCIV. F from a given point a straight line be drawn touching a circle given in pofition; the straight line is given in pofition and magnitude. Let the straight line AB be drawn from the given point A touching the circle BC given in pofition; AB is given in pofition and magnitude. Take D the center of the circle, and join DA, DB. because each of the points D, A is given, the straight B a. 29. Dat. line AD is given in pofition and magni- C d. 6. Def. that circle is therefore given in pofition. and the circle BC is given in pofition, there e. 28. Dat. fore the point B is given. the point A is alfo given; therefore the ftraight line AB is given in pofition and magnitude. 92. 2. 17.3. IF PROP. XCV. Fa ftraight line be drawn from a given point without a circle given in pofition; the rectangle contained by the fegments betwixt the point and the circumference of the circle is given. Let the straight line ABC be drawn from the given point A with out the circle BCD given in pofition, cut ting it in B, C; the rectangle BA, AC is From the point A draw AD touching C b. 94. Dat. the circle; therefore AD is given b in pofition and magnitude. and because AD is which is e. 56. Dat. given, the fquare of AD is given d. 36. 3. equal to the rectangle BA, AC. ВА therefore the rectangle BA, AC is given. PROP. IF PROP. XCVI. Fa ftraight line be drawn thro' a given point within a circle given in pofition, the rectangle contained by the fegments betwixt the point and the circumference of the circle is given. Let the straight line BAC be drawn thro' the given point A within the circle BCE given in pofition; the rectangle BA, AC is given. Take D the center of the circle, join AD and produce it to the points E, F. because D 2) E C c 93. 2. 29. Dat. b. 28. Dat. and the rectangle EA, AF is therefore given; and it is equal to c. 35. 3. the rectangle BA, AC which confequently is given. PROP. XCVII. a ftraight line be drawn within a circle given in magnitude cutting off a fegment containing a given angle; if the angle in the fegment be bifected by a ftraight line produced till it meets the circumference, the straight lines which contain the given angle fhall both of them together have a given ratio to the straight line which bifects the angle. and the rectangle contained by both thefe lines together which contain the given angle, and the part of the bifecting line cut off below the base of the fegment, shall be given. Let the straight line BC be drawn within the circle ABC given in magnitude cutting off a fegment containing the given angle BAC, and let the angle BAC be bifected by the F straight line AD; BA together with AC has a given ratio to AD; and the rectangle contained by BA and AC together, and the ftraight line ED cut off from AD below BC the bafe of the fegment, is gi ven. BE 94. Join BD; and because BC is drawn within the circle ABC giveri in magnitude cutting off the fegment BAC containing the given a. 91. Dat. angle BAC; BC is given in magnitude. by the fame reafon BD b. 1. Dat. is given; therefore the ratio of BC to BD is given. the angle BAC is bifected by AD, as BA to AC, fo is and, by permutation, as AB to BE, fo is AC to CE; wherefore & as BA and AC together to BC, fo is AC to CE. c. 3. 6. d. 12. s. and because BE to EC; and because the angle BAE is equal to EAC, and the F e. 21. 3. angle ACE to ADB; the triangle ACE" is equiangular to the triangle ADB; therefore as AC to CE, fo is AD to DB. but as AC to CE, fo is BA together with AC to BC; as therefore BA and AC to. BC, fo is AD to DB; and, by permutation, as BA and AC to AD, fo is BC to BD. and the ratio of BC to BD is given, therefore the ratio of BA together with AC to AD is given. E B D Alfo the rectangle contained by BA and AC together, and DE is given. Becaufe the triangle BDE is equiangular to the triangle ACE, as ED to DE, fo is AC to CE; and as AC to CE, fo is BA and AC to BC; therefore as BA and AC to BC, fo is BD to DE. wherefore the rectangle contained by BA and AC together, and DE is equal to the rectangle CB, BD. but CB, BD is given; therefore the rectangle contained by BA and AC together, and DE is given. Otherwife. Produce CA and make AF equal to AB, and join BF. and bea. 5. and 32. caufe the angle BAC is double of each of the angles BFA, BAD, I. a the angle BFC is equal to BAD; and the angle BCA is equal to BDA, therefore the triangle FCB is equiangular to ADB. as therefore FC to CB, fo is AD to DB, and, by permutation, as FC, that is BA and AC together to AD, fo is CB to BD. and the ratio of CB to BD is given, therefore the ratio of BA and AC to AD is given. And because the angle BFC is equal to the angle DAC, that is to the angle DBC, and the angle ACB equal to the angle ADB; the triangle FCB is equiangular to BDE, as therefore FC to CB, fo is BD to DE; therefore the rectangle contained by FC, that is BA and AC together, and DE is equal to the rectangle CB, BD which is given, and therefore the rectangle contained by BA, AC together, and DE is given. PROP. a Fa ftraight line be drawn within a circle given in magnitude cutting off a fegment containing a given angle; if the angle adjacent to the angle in the fegment be bifected by a straight line produced till it meet the circumference again and the bafe of the fegment; the excess of the straight lines which contain the given angle fhall have a given ratio to the fegment of the bifecting line which is within the circle; and the rectangle contained by the fame excefs and the fegment of the bifecting line betwixt the bafe produced and the point where it again meets the circumference, fhall be given. Let the ftraight line BC be drawn within the circle ABC given in magnitude cutting off a fegment containing the given angle BAC, and let the angle CAF adjacent to BAC be bifected by the straight line DAE meeting the circumference again in D, and BC the base of the segment produced in E; the excess of BA, AC has a given ratio to AD; and the rectangle which is contained by the fame excefs and the ftraight line ED, is given. Join BD, and thro' B draw BG parallel to DE meeting AC produced in G. and becaufe BC cuts off from the cir cle ABC given in magnitude the fegment BAC contain ing a given angle, BC is therefore gi A a. 91. Das. C G CAE is equal to the alternate angle AGB, and EAF to the interior and oppofite angle ABG; therefore the angle AGB is equal to ABG, and the ftraight line AB equal to AG; fo that GC is the excefs of BA, AC. and because the angle BGC is equal to GAE, that is to EAF, or the angle BAD; and that the angle BCG is equal to the oppofite interior angle BDA of the quadrilateral BCAD in the circle; therefore the triangle BGC is equi angular to BDA. therefore as GC to CB, fo is AD to DB, and, by permutation, as GC, which is the excefs of BA, AC to AD, fo is CB to BD. and the ratio of CB to BD is given; therefore the ratio of the ex- D F A 95. And because the angle GBC is equal angle BCG is equiangular to BDE. C E G DE, and confequently the rectangle GC, DE is equal to the rect angle CB, BD which is given, because its fides CB, BD are given. therefore the rectangle contained by the excess of BA, AC and the ftraight line DE is given. IF PROP. XCIX. F from a given point in the diameter of a circle given in pofition, or in the diameter produced, a ftraight line be drawn to any point in the circumference, and from that point a ftraight line be drawn at right angles to the first, and from the point in which this meets the circumference again, a straight line be drawn parallel to the first; the point in which this parallel meets the diameter is given; and the rectangle contained by the two parallels is given. In BC the diameter of the circle ABC given in position, or in BC produced, let the given point D be taken, and from D let a ftraight line DA be drawn to any point A in the circumference, and let AE be drawn at right angles to DA, and from the point E where it meets the circumference again let EF be drawn parallel to DA meeting BC in F; the point F is given, as alfo the rectangle AD, EF. Produce EF to the circumference in G, and join AG. becaufe a. Cor. 5.4.GEA is a right angle, the straight line AG is the diameter of the circle ABC; and BC is also a diameter of it; therefore the point H where they meet is the center of the circle, and confequently H is given. and the point D is given, wherefore DH is given in magni tude. |