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ven in pofition, in a given angle HGC, for it is equal a to the a 29. 1. given angle ADC; AG is given in position b; but it is given b 32. dat. also in magnitude, because it is equal to CAD which is given in c 34. 1. magnitude; therefore because G one of the extremities of the ftraight line GH given in pofition and magnitude is given, the other extremity H is given d; and the straight line EAF, which d 30. dat. is drawn through the given point H parallel to BC given in pofition, is therefore given in position.
€ 31. dat.
PRO P. XXXVIII.
IF a straight line be drawn from a given point to two
parallel straight lines given in polition ; the ratio of the segments between the given point and the parallels Thall 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 pofition, in a given angle EKC ; EK is
A FH B
с KG D CG K D given in position a ; and AB, CD are given in position ; there a 33. dat. fore b the points H, K are given : And the point E is given ; b 28. data wherefore c EH, EK are given in magnitude, and the rario ofc 29. dat, them is therefore given. But as EH 10 EK, fo is EF to EG, be.d 1. dat. cause AB, CD are parallels ; therefore the ratio of EF to EG is given.
PRO P. XXXIX.
IF the ratio of the segments of a straight line between See N.
a given point in it and two parallei Atraight lines, be given ; if one of the parallels be given in position, the other is also given in polition,
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 pofition.
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
1 33. dat. given angle AHD; AH is given a in
А position; and BC is likewise given in
position, therefore the point # is gi-B D/HC b 28. dat. ven b:. The point A is also given ; € 29. dat. wherefore AH is given in magnitude c,
and, because FG, BC are parallels,
wherefore the ratio of AK to AH is given ; but AH is given d 2. dat. in magnitude, therefore d AK is given in magnitude ; and it is
; € 30. dat. also given in position, and the point A is given; wherefore e
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 pof31. dat. fition, therefore f FG is given in position.
See N. 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 also is 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 fegments GE, GF into which the straight line GEF is cut by the three parallels, be given ; the third parallel HK is given in pofition.
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 gi. ven angle LMD; LM is given in position a ; and CD is given a 33. dat. in position, wherefore the point M is given b; and the point Lb 28. dat. is given, LM is therefore given in magnitude c; and because c 29. dat. the ratio of GE to GF is given, and as GE to GF, so is NL to H GN
E L B
E L BH
Cor. 6. or
M D NM; the ration of NL to NM is given; and therefore d the ratio of ML to LN is given ; but LM is given in magnitude, wherefore e LN is given in magnitude ; and it is also given in pofition, and the point L is given ; wherefore f the point N is f 30. dat. given ; and because the straight line HK is drawn through the given point N parallel to CD which is given in position, therefore HK is given in position &.
e 2. dat.
8 31. date
P R O P. XLI.
IF a straight line meets three parallel straight lines See N.
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.
In AB take a given point L, and draw LM perpendicular to CD, meet.
А G L B ing EF in N; therefore a LM is given
a 33; dar. in pofition ; and CD, EF are given C_H M D in position, wherefore the points M, Nare given : And the point L is given; therefore bthe straight lines LM, MN
b 29. date are given in magnitude; and the ratio E K N F
e. 1. dat. of LM to MN is therefore given c: But as LM to MN, so i
GH to HK; wherefore the ratio of GH to HK is given.
IF each of the sides of a triangle be given in magni
tude; the triangle is given in species.
Let each of the sides of the triangle ABC be given in mag
nitude ; the triangle ABC is given in fpecies. 22. 1. Make a triangle a DEF the fides of which are cqual, cach
to each, to the given straight lines AB, BC, CA ; which can
F the base BC; the angle b 8. 5. EDF is equal b to the angle BAC; therefore, because the angle
, EDF, which is equal to the angle BAC, has been found, the c. 1. def. angle BAC is given ç, in like manner the angles at B, C are
given. And because the sides AB, BC, CA are given, their d. 1. dat: ratios to one another are given d, therefore the triangle ABC def.
is given e in fpecies.
tude; the triangle is given in fpecies.
Let each of the angles of the triangle ABC be given in mago nitude ; the triangle ABC is given in fpecies. Take a straight line DE given in
A position and magnitude, and at the
D 2 23. 1. points D, E make a the angle EDF
equal to the angle BAC, and the
ven, wherefore each of those at the points D, E, F is given : And because the straight line FD is drawn to the given point D in DE which is given in position, making the given angle EDF; therefore DF is given in posicion b. In like manner EF b 32. dat. also is given in position; wherefore the point F is given : And the points D, E are given; therefore each of the straight lines DE, EF, FD is given c in magnitude ; wherefore the triangle c 29. dat. DEF is given in fpecies d; and it is fimilar c to the triangled 42. dat. ABC; which therefore is given in species.
PRO P. XLIV.
sides about it have a given ratio to one another; the triangle is given in species.
a 32. dati
Let the triangle ABC have one of its angles BAC given, and let the fides BA, AC about it have a given ratio to one another; the triangle ABC is given in specics. Take a straight line DE given in position and magnitude,
a and at the point D in the given straight line DE, make the angle EDF cqual to the given angle BAC; wherefore the angie EDF is given ; and because the straight line FD is drawn to the given point D in ED which is given in position, making the given angle EDF; therefore FD is given in position a. And because the ratio of BA to AC is given,
D make the ratio of ED to DF the fame with it, and join EF; and be. cause the ratio of ED to DF is gi. B CE
E F ven, and £D is given, therefore b DF is given in magnitude ; b 2. dat. and it is given also in position, and the point D is given, wherefore the point F is given c; and the points D, E are given, 30. dat. wherefore DE, EF, FD are given din magnitude; and the d 29. dat. triangle DEF is therefore given e in fpecies; and because the c 42. dat. triangles ABC, DEF have one angle BAC equal to one angle EDF, and the Gdes about these angles proportionals; the triangies are f fimilar; but the triangle DÈF is given in species, i 6. 6. and therefore also the triangle ABC.