line G, that is, EF, cannot be less than HK: and if G be equal to HK, EF also is equal to it; wherefore EF is at right angles to CD: for if it be not, EF would be greater than HK, which is absurd. Therefore the angle EFD is a right, and consequently a given angle. But if the straight line G be not equal to HK, it must be greater than it: produce HK, and take HL, equal to G; and from the centre H, at the distance HI, describe the circle MLN, and join HM, HN: and because the circle (6. def.) MLN, and the straight line CD, are given in position, the points M, N are (28. dat.) given: and the point His given, wherefore the straight A E H B lines HM, HN, are given in position (29. dat.): and CD is. given in position: therefore the K angles HMN,HNM, are given CF in position (A. def.): of the straight lines HM, HN,let HN be that which is not parallel to G EF, for EF cannot be parallel to both of them; and draw EO parallel to HN: EO therefore is equal (34. 1.) to HN, that is to G; and EF is equal to G, wherefore EO is equal to EF, and the angle EFO to the angle EOF, that is (29. 1.), to the given angle HNM; and because the angle HNM, which is equal to the angle EFO, or EFD, has been found: therefore the angle EFD, that is the angle AEF, is given in magnitude (1. def.); and consequently the angle EFC. Ir a straight line given in magnitude be drawn from a point to a straight line given in position, in a given angle; the straight line drawn through that point parallel to the straight line given in position, is given in position.* A H F Let the straight line AD given in magnitude be drawn from the point A to the straight line BC given in position, in the given angle ADC: the E straight line EAF drawn through a parallel to BC is given in position. In BC take a given point G, and draw GH parallel to AD: and because HG is drawn to a given point G in the straight * See Note. B D G 1 line BC given in position, in a given angle HGC, for it is equal (29. 1.) to the given angle ADC; HG is given in position (32. dat.); but it is given also in magnitude, because it is equal to (34, 1.) AD which is given in magnitude; therefore because G one of the extremities of the straight line GH given in position and magnitude is given, the other extremity H is given (30. dat.); and the straight line EAF, which is drawn through the given point H parallel to BC given in position, is therefore given (31. dat.) in position. Ir a straight line be drawn from a given point to two parallel straight lines given in position, the ratio of the segments between the given point and the parallels shall be given. Let the straight line EFG be drawn from the given point E to the parallels AB, CD, the ratio of EF to EG is given. From the point E draw EHK perpendicular to CD; and because from a given point E the straight line EK is drawn to CD which is given in position, in a given angle EKC; EK is given in position (33. dat.); and AB, CD are given in position: therefore (28. dat.) the points H, K are given; and the point E is given; wherefore (29. dat.) EH, EK are given in magnitude, and the ratio (1. dat.) of them is therefore given. But as EH to EK, so is EF to EG, because AB, CD are parallels therefore the ratio of EF to EG is given. olistne d95PROP. XXXIX. 35, 36. If the ratio of the segments of a straight line between a given point in it and two parallel straight lines be given, if one of the parallels be given in position, the other is also given in position. From the given point A, let the straight line AED be drawn to the two parallel straight lines FG, BC, and let the ratio of the segments AE, AD be given; if one of the parallels BC be given in position, the other FG is also given in position. From the point A, draw AH perpendicular to BC, and let it meet FG in K: and because AH is drawn from the given point A to the straight line BC given in position, and makes a given A B D H e E K angle AHD; AH is given (33. dat.) in position; and BC is likewise given in position: therefore the point H is given (28. dat.): the point A is also given; wherefore AH is given in magnitude (29. dat.) and because FG,BC are parallels, as AE to AD, so is AK to AH; and the ratio F of AE to AD is given, wherefore the ratio of AK to AH is given; but AH is given in magnitude, therefore (2. dat.) AK is given in magnitude; and it is also given in position, and the point A is given; wherefore (30. dat.) the point K is given. And because the straight line FG is drawn through the given point K parallel to BC which is given in position, therefore (31. dat.) FG is given in position. If the ratio of the segments of a straight line into which it is cut by three parallel straight lines, be given; if two of the parallels are given in position, the third is also given in position.* Let AB, CD, HK be three parallel straight lines of which AB, CD are given in position; and let the ratio of the seg * See Note. ments GE, GF into which the straight line GEF is cut by the three parallels, be given; the third parallel HK is given in po sition. In AB take a given point L, and draw LM perpendicular to CD, meeting HK in N; because LM is drawn from the given point L to CD which is given in position, and makes a given angle LMD to LM is given in position (33. dat.); and CD is given in position, wherefore the point M is given (28. dat.); and the point Lis given, LM is therefore given in magnitude (29. dat.); and because the ratio of GE to GF is given, and as GE to GF, so is NL to NM; the ratio of NL to NM is given; and therefore (cor. 6. or 7. dat.) the ratio of ML to LN is given; but LM is given in magnitude (cor. 6. or 7. dat.), wherefore (2. dat.) LN is given in magnitude; and it is also given in position, and the point Lis given, wherefore (30.dat.) the point N is given; and because the straight line HK is drawn through the given point N parallel to CĎ which is given in position, therefore HK is given in position (31. dat.). Ir a straight line meets three parallel straight lines which are given in position, the segments into which they cut it have a given ratio. Let the parallel straight lines AB, CD, EF, given in position, be cut by the straight line GHK; the ratio of GH to HK is given. C G L B H In AB take a given point L, and A draw LM perpendicular to CD, meeting EF in N; therefore (33. dat.) LM is given in position; and CD, EF are given in position, wherefore the points M, N are given; and the point L is given; therefore (29. dat.) the straight E K lines LM, MN are given in magni tude; and the ratio of LM to MN is therefore given (1. dat.): but as LM to MN, so is GH to HK; wherefore the ratio of GH to HK is given. If each of the sides of a triangle be given in magnitude, the triangle is given in species. Let each of the sides of the triangle ABC be given in magnitude, the triangle ABC is given in species. A D Make a triangle (22. 1.) DEF, the sides of which are equal, each to each, to the given straight lines AB, BC, CA, which can be done; because any two of them must be greater than the third; and let DE be equal to AB, EF to BC, and FD to CA; and because the two sides ED, DF are equal to the two BA, AC, each to each, and the base EF equal to the base BC; B the angle EDF, is equal (8. 1.) CE F to the angle BAC; therefore, because the angle EDF, which is equal to the angle BAC, has been found, the angle BAC is given (1. def.); in like manner the angles at B, C are given. And because the sides AB, BC, CA are given, their ratios to one another are given (1. dat.), therefore the triangle ABC, is given (3. def.) in species. If each of the angles of a triangle be given in magnitude, the triangle is given in species. A Let each of the angles of the triangle ABC be given in magnitude, the triangle ABC is given in species. Take a straight line DE given in position and magnitude, and at the points D, E make (23. 1.) the angle EDF equal to the angle BAC; and the angle DEF equal to ABC; therefore the other angles EFD, BCA are B D 44 CE F equal, and each of the angles at the points A, B, C, is given |