PROP. LXXX. IF the sides about an angle of a triangle have a given ratio to one another, and if the perpendicular drawn from that angle to the base has a given ratio to the base; the triangle is given in species. Let the sides BA, AC, about the angle BAC of the triangle ABC have a given ratio to one another, and let the perpendicular AD have a given ratio to the base BC; the triangle ABC is given in species. 1 A First, let the sides AB, AC be equal to one another, therefore the perpendicular AD bisects a the base BC; and the ratio of AD to BC, and therefore to its half DB, is given; and the angle ADB is given; wherefore the triangle * ABD, and consequently the triangle ABC, is given b in species. B 2 A D * 43. dat. C b44. dat. Ad 6. dat. But let the sides be unequal, and BA be greater than AC; and make the angle CAE equal to the angle ABC; because the angle AEB is common to the triangles AEB, CEA, they are similar; therefore as AB to BE, so is CA to AE, and, by permutation, as BA to AC, so is BE to EA, and so is EA to EC; and the ratio of BA to AC is given, therefore the ratio of BE to EA, and the ratio of EA to EC, as also the ratio of BE to EC is given ; wherefore the ratio of EB to c 9. dat. BC is given; and the ratio of AD to BC is given by the hypothesis, therefore the ratio of AD to BE is given; and the ratio BE to EA was shown to be given; wherefore the ratio of AD to AE is given, and ADE is a right angle, therefore the triangle ADE is given in species, and the angle AEB given; the ra-e 46. dat. tio of BE to EA is likewise given, therefore b the triangle ABE is given in species, and consequently the angle EAB, as also the angle ABE, that is, the angle CAE, is given; therefore the angle BAC is given, and the angle ABC being also given, the triangle ABC is given in species. B FCE D How to find a triangle which shall have the things which are mentioned to be given in the proposition, is evident in the first case; and to find it the more easily in the other case, it is to be observed that, if the straight line EF equal to f43. dat. FA be placed in EB towards B, the point F divides the base BC into the segments BF, FC which have to one another the ratio of the sides BA, AC, because BE, EA or EF, and 19. 5. EC were shown to be proportionals, therefore BF is to FC as BE to EF, or EA, that is, as BA to AC; and AE cannot be less than the altitude of the triangle ABC, but it may be equal to it, which, if it be, the triangle, in this case, as also the ratio of the sides, may be thus found; having given the ratio of the perpendicular to the base, take the straight line GM given in position and magnitude, for the base of the triangle to be found; and let the given ratio of the perpendicular to the base be that of the straight line K to GH, that is, let K be equal to the perpendicular; and suppose GLH to be the triangle which is to be found, therefore having made the angle HLM equal to LGH, it is required that LM be perpendicular to GM, and equal to K; and because GM, ML, MH are proportionals, as was shown of BE, EA, EC, the rectangle GMH is equal to the square of ML. Add the common square of NH, (having bisected GH in N), and the square of NM is equals to the squares of the given straight lines NH and ML or K; therefore the square of NM and its side NM, is given, as also the point M, viz. by taking the straight line NM, the square of which is equal to the squares of NH, ML. Draw ML equal to K, at right angles to GM; and because ML is given in position and magnitude, therefore the point L is given, join LG, LH; then the triangle LGH is that which was to be found, for the square of NM is equal to the squares of NH and ML, and taking away the common square of NH, the rectangle GMH is equal to the square of ML therefore as GM to ML, so is ML to MH, and the triangle LGM is therefore equiangular to HLM, and the angle HLM equal to G g 6. 2. h 6. 6. 3 K L R NQ H M P the angle LGM, and the straight line LM, drawn from the vertex of the triangle making the angle HLM equal to LGH, is perpendicular to the base and equal to the given straight line K, as was required; and the ratio of the sides GL, LH is the same with the ratio of GM to ML, that is, with the ratio of the straight line which is made up of GN the half of the given base and of NM, the square of which is equal to the squares of GN and K, to the straight line K. And whether this ratio of GM to ML is greater or less than the ratio of the sides of any other triangle upon the base GH, and of which the altitude is equal to the straight line K, that is, the vertex of which is in the parallel to GH drawn through the point L, may be thus found. Let OGH be any such triangle, and draw OP, making the angle HOP equal to the angle OGH; therefore, as before, GP, PO, PH are proportionals, and PO cannot be equal to LM, because the rect= angle GPH would be equal to the rectangle GMH, which is impossible; for the point P cannot fall upon M, because O would then fall on L; nor can PO be less than LM, therefore it is greater; and consequently the rectangle GPH is greater than the rectangle GMH, and the straight line GP greater than GM: therefore the ratio of GM to MH is greater than the ratio of GP to PH, and the ratio of the square of GM to the square of ML is therefore i greater than the ratio of the i2 Cor. square of GP to the square of PO, and the ratio of the straight 20. & line GM to ML, greater than the ratio of GP to PO. But as GM to ML, so is GL to LH; and as GP to PO, so is GO to OH; therefore the ratio of GL to LH is greater than the ratio of GO to OH; wherefore the ratio of GL to LH is the greatest of all others; and consequently the given ratio of the greater side to the less, must not be greater than this ratio. But if the ratio of the sides be not the same with this greatest ratio of GM to ML, it must necessarily be less than it : let any less ratio be given, and the same things being supposed, viz. that GH is the base, and K equal to the altitude of the triangle, it may be found as follows. Divide GH in the : point Q, so that the ratio of GQ to QH may be the same with the given ratio of the sides; and as GQ to QH, so make GP to PQ, and so will f PQ be to PH; wherefore the square f 19.5. of GP is to the square of PQ, asi the straight line GP to PH and because GM, ML, MH are proportionals, the square of GM is to the square of ML, asi the straight line GM to MH: but the ratio of GQ to QH, that is, the ratio GP to PQ, is less than the ratio of GM to ML; and therefore the ratio of the square of GP to the square of PQ is less than the ratio of the square of GM to that of ML; and consequently the ratio of the straight line GP to PH is less than the ratio of GM to MH; and, by division, the ratio of GH to HP is less than that of GH to HM; wherefore the straight line HP is k 10.5. greater than HM, and the rectangle GPH, that is, the square of PQ, greater than the rectangle GMH, that is, than the square of ML, and the straight line PQ is therefore greater than ML. Draw LR parallel to GP: and from P draw PR at right angles to GP: because PQ is greater than ML or PR, the circle described from the centre P, at the distance PQ, must necessarily cut LR in two points; let these be O, S, and join OG, OH; SG, SII: each of the triangles OGH, SGH have the things mentioned to be given in the proposition: join OP, SP; and because as GP to PQ, or PO, so is PO to PH, the triangle OGP is equiangular to HOP; as, therefore OG to GP, so is HO to OP; and, by permutation, as GO to OH so is GP to PO, or PQ: and so is GQ to QH : therefore the triangle OGH has the ratio of its sides GO, OH the same with the given ratio of GQ to QH : and the perpendicular has to the base the given ratio of K to GH, because the perpendicular is equal to LM or K: the like may be shown in the same way of the triangle SGH. This construction by which the triangle OGH is found, is shorter than that which would be deduced from the demonstra tion of the datum, by reason that the base GH is given in position and magnitude, which was not supposed in the demonstration the same thing is to be observed in the next proposition. M. PROP. LXXXI. a 4.6. IF the sides about an angle of a triangle be unequal and have a given ratio to one another, and if the perpen. dicular from that angle to the base divides it into segments that have a given ratio to one another, the trian gle is given in species. Let ABC be a triangle, the sides of which about the angle BAC are unequal, and have a given ratio to one another, and let the perpendicular AD to the base BC divide it into the segments BD, DC, which have a given ratio to one another, the triangle ABC is given in species. Let AB be greater than AC, and make the angle CAE equal to the angle ABC; and because the angle AEB is common to the triangles ABE, CAE, they are a equiangular to one another: therefore as AB to BE, so is CA to AE, and, AC, so BE to by permutation, as AB to to EA was shown to be given, therefore b the ratio of DE to EA 6. dat. d 7. dat. is given and ADE is a right angle, wherefore the triangle e 46. dat. ADE is given in species, and the angle AED given: and the ratio of CE to EA is given, therefore the triangle AEC is gi, f 44. dat. ven in species, and consequently the angle ACE is given, as also the adjacent angle ACB. In the same manner, because the ratio of BE to EA is given, the triangle BEA is given in species, and the angle ABE is therefore given: and the angle ACB is given; wherefore the triangle ABC is given & in spe- g 45. dat. cies. But the ratio of the greater side BA to the other AC mustbe less than the ratio of the greater segment BD to DC: because the square of BA is to the square of AC, as the squares of BD, DA to the squares of DC, DA; and the squares of BD, DA have to the squares of DC, DA a less ratio than the square of BD has to the square of DC †, because the square of BD is greater than the square of DC; therefore the square of BA has to the square of AC a less ratio than the square of BD has to that of DC: and consequently the ratio of BA to AC is less than the ratio of BD to DC. This being premised, a triangle which shall have the things mentioned to be given in the proposition, and to which the triangle ABC is similar, may be found thus: take a straight line GH given in position and magnitude, and divide it in K, so that the ratio of GK to KH may be the same with the given ratio of BA to AC; divide also GH in L, so that the ratio of GL to LH may be the same with the given ratio of BD to If A be greater then B, and C any third magnitude; then A and C toge. ther have to B and C together a less ratio than A has to B. Let A be to B as C to D, and because A is greater than B, C is greater than D: but as A is to B, so A and C to B and D; and A and C have to B and C a less ratio than A and C have to B and D, because C is greater than D, therefore A and C have to B and C a less ratio than A to B. |