tion and magnitude, and at the point G in GH make the angle HGK equal to the given angle CAD, and take GK equal to GH; join KH, and draw GL perpendicular to it: then the ratio of HK to the half of GL is the same with the ratio of the rectangle DC, CE to the triangle ABC: because the angles HGK, DAC, at the vertices of the isosceles triangles GHK, ADC, are equal to one another, these triangles are similar; and because GL, AF are perpendicular to the bases HK, DC, as HK to GL, so is (4. 6. 22. 5.) (DC to AF, and so is) the rectangle DC, CE to the rectangle AF, CE; but as GL to its half, so is the rectangle AF, CE to its half, which is the triangle ACE, or the triangle ABC; therefore, ex æquali, HK is to the half of the straight line GL, as the rectangle DC, CE is to the triangle ABC. Cor. And if a triangle have a given angle, the space by which the square of the straight line which is the difference of the sides which contain the given angle is less than the square of the third side, shall have a given ratio to the triangle. This is demonstrated the same way as the preceding proposition, by help of the second case of the lemma. PROP. LXXVII. L. If the perpendicular drawn from a given angle of a triangle to the opposite side, or base, have a given ratio to the base, the triangle is given in species. * Let the triangle ABC have the given angle BAC, and let the perpendicular AD drawn to the base BC have a given ratio to it; the triangle ABC is given in species. If ABC be an isosceles triangle, it is evident (5. & 32. 1.) that B RDC EO H M F if any one of its angles be given, the rest are also given ; and therefore the triangle is given in species, without the consideration of the ratio of the perpendicular to the base, which in this case is given by prop. 50. But when ABC is not an isosceles triangle, take any straight line EF given in position and magnitude, and upon it describe . See Note. the segment of a circle EGF containing an angle equal to the given angle BAC; draw GH bisecting EF at right angles, and join EG, GF: then, since the angle EGF is equal to the angle BAC, and that EGF is an isosceles triangle, and ABC is not, the angle FEG is not equal to the angle CBA: draw EL, making the angle FEL equal to the angle CBA; join FL, and draw LM perpendicular to EF; then, because the triangles ELF, BAC are equiangular, as also are the triangles MLE, DAB, as ML to LE, so is DA to AB ; and as LE to EF, so is AB to BC; wherefore, ex æquali, as LM to EF, so is AD to BC ; and because the ratio of AD to BC is given, therefore the ratio of LM to EF is given ; and EF is given, wherefore (2. dat.) LM also is given. Complete the parallelogram LMFK; and because LM is given, FK is given in magnitude ; it is also given in position, and the point F is given, and consequently (30. dat.) the point K; and because through K the straight line KL is drawn parallel to EF which is given in position, therefore (31. dat.) KL is given in position: and the circumference ELF is given in position; therefore the point L is given (28. dat.). And because the points L, E, F, are given, the straight lines LE, EF, FL, are given (29. dat.) in magnitude ; therefore the triangle LEF is given in species (42. dat.); and the triangle ABC is similar to LEF, wherefore also ABC is given in species. Because LM is less than GH, the ratio of LM to EF, that is, the given ratio of AD to BC, must be less than the ratio of GH to EF, which the straight line, in a segment of a circle containing an angle equal to the given angle, that bisects the base of the segment at right angles, has unto the base. COR. 1. If two triangles, ABC, LEF, have one angle BAC equal to one angle ELF, and if the perpendicular AD be to the base BC, as the perpendicular LM to the base EF, the triangles ABC, LEF are similar. Describe the circle EFG about the triangle ELF, and draw LN parallel to EF; join EN, NF, and draw NO perpendicular to EF ; because the angles ENF, ELF are equal, and that the angle EFN is equal to the alternate angle FNL, that is, to the angle FEL in the same segment; therefore the triangle NEF is similar to LEF; and in the segment EGF there can be no other triangle upon the base EF, which has the ratio of its perpendicular to that base the same with the ratio of LM or NO to EF, because the perpendicular must be greater or less than LM or NO; but, as has been shown in the preceding demonstration, a triangle similar to ABC can be described in the segment EGF upon the base EF, and the ratio of its perpendicular to the base is the same, as was there shown, with the ratio of AD to BC, that is, of LM to EF ; therefore that triangle must be either LEF, or NEF, which therefore are similar to the triangle ABC. Cor. 2. If a triangle ABC have a given angle BAC, and if the straight line AR drawn from the given angle to the opposite side BC, in a given angle ARC, have a given ratio to BC, the triangle ABC is given in species. Draw AD perpendicular to BC; therefore the triangle ARD is given in species ; wherefore the ratio of AD to AR is given : and the ratio of AR to BC is given, and consequently (9. dat.) the ratio of AD to BC is given ; and the triangle ABC is therefore given in species (77. dat.). CoR. 3. If two triangles ABC, LEF have one angle BAC equal to one angle ELF, and if straight lines drawn from these angles to the bases, making with them given and equal angles, have the same ratio to the bases, each to each ; then the triangles are similar ; for having drawn perpendiculars to the bases from the equal angles, as one perpendicular is to its base, so is the other to its base (4. 6. 22. 5.); wherefore, by Cor. 1. the triangles are similar. A triangle similar to ABC may be found thus : having described the segment EGF, and drawn the straight line GH, as was directed in the proposition, find FK, which has to EF the given ratio of AD to BC; and place FK at right angles to EF from the point F; then, because, as has been shown, the ratio of AD to BC, that is of FK to EF, must be less than the ratio of GH to EF; therefore FK is less than GH; and consequently the parallel to EF, drawn through the point K, must meet the circumference of the segment in two points : let L be either of them, and join EL, LF, and draw LM perpendicular to EF: then, because the angle BAC is equal to the angle ELF, and that AD is to BC, as KF, that is, LM to EF, the triangle ABC is similar to the triangle LEF, by Cor. 1. 80. PROP. LXXVIII. If a triangle have one angle given, and if the ratio of the rectangle of the sides which contain the given angle to the square of the third side be given, the triangle is given in species. Let the triangle ABC have the given angle BAC, and let the ratio of the rectangle BA, AC to the square of BC be given ; the triangle ABC is given in species. From the point A, draw AD perpendicular to BC; the rectangle AD, BC has a given ratio to its half (41. 1.), the triangle ABC ; and because the angle BAC is given, the ratio of the triangle ABC to the rectangle BA, AC is given (Cor. 62. dat.); and by the hypothesis, the ratio of the rectangle BA, AC to the square of BC is given; therefore (9. dat.) the ratio of the rectangle AD, BC to the square of BC, that is (1. 6.), the ratio of the straight line AD to BC is given ; wherefore the triangle ABC is given in species (77. dat.). A triangle similar to ABC may be found thus : take a straight line EF given in position and magnitude, and make the angle FEG equal to the given angle BAC, and draw FH perpendicular to EG, and BK perpendicular to AC: therefore the triangle ABK, EFH are M 0 A similar, and the rectangle E L P G Let the given ratio of the rectangle BA, AC to the square of BC be the same with the ratio of the straight line EF to FL ; therefore, ex æquali, the ratio of the rectangle AD, BC to the square of BC, that is, the ratio of the straight line AD to BC, is the same with the ratio of HF to FL; and because AD is not greater than the straight line MN in the segmentof the circle described about the triangle ABC, which bisects BC at right angles; the ratio of AD to BC, that is, of HF to FL, must not be greater than the ratio of MN to BC : let it be so; and, by the 77th dat. find a triangle OPQ which has one of its angles POQ equal to the given angle BAC, and the ratio of the perpendicular OR, drawn from that angle to the base PQ the same with the ratio of HF to FL ; then the triangle ABC is R similar to OPQ : because, as has been shown, the ratio of AD to BC is the same with the ratio of (HF to FL, that is, by the construction, with the ratio of) OR to PQ ; and the angle BAC is equal to the angle POQ; therefore the triangle ABC is similar , (1. Cor. 77. dat.) to the triangle POQ. Otherwise, Let the triangle ABC have the given angle BAC, and let the ratio of the rectangle BA, AC to the square of BC be given ; the triangle ABC is given in species. Because the angle BAC is given, the excess of the square of both the sides BA, AC together above the square of the third side BC has a given (76. dat.) ratio to the triangle ABC. Let the figure D be equal to this excess; therefore the ratio of D to the triangle ABC is given: and the ratio of the triangle ABC to the rectangle BA, AC is given (Cor. 62. dat.), because BAC is a given angle; and the rectangle BA, AC has a given ratio to the square of BC: wherefore (10. dat.) the ratio of D to the D square of BC is given ; and by composition (7. dat.) the ratio of the space D to B С gether with the square of BC to the square of BC is given; but D together with the square of BC is equal to the square of both BA and AC together; therefore the ratio of the square of BA, AC together to the square of BC is given ; and the ratio of BA, AC together to BC is therefore given (59. dat.); and the angle BAC is given, wherefore (48. dat.) the triangle ABC is given in species. The composition of this, which depends upon those of the 76th and 48th propositions, is more complex than the preceding composition, which depends upon that of prop. 77, which is easy. PROP. LXXIX. K. Ir a triangle have a given angle, and if the straight line drawn from that angle to the base, making a given angle with it, divide the base into segments which have a given ratio to one another; the triangle is given in species. * Let the triangle ABC have the given angle BAC, and let the straight line AD drawn to the base BC making the given angle ADB, divide BC into the segments BD, DC which have See Note. |