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* 30. 1.
two straight lines KH, HM upon the opposite sides of it make the adjacent angles equal to two right angles, KH is in the same straight line* with HM: * 14. 1. and because the straight line HG meets the parallels KM, FG, the alternate angles MHG, HGF* are * 29. 1. equal, add to each of these the angle HGL: therefore the angles MHG, HGL, are equal + to the angles + 2 Ax. HGF, HGL: but the angles MHG, HGL, are equal* to two right angles; wherefore also the angles29. 1. HGF, HGL are equalt to two right angles, and there- +1 Ax. fore FG is in the same straight linet with GL: and + 14. 1. because K F is parallel to HG+, and HG to ML; KF is parallel* to ML: and KM, FL aret parallel-;
+ Constr. wherefore KFLM ist a parallelogram: and because
+ Def. 34. the triangle ABD is equal to the parallelogram HFt, 1. and the triangle DBC to the parallelogram GM; the + Constr. whole rectilineal figure ABCD is equalt to the whole + 2 Ax. parallelogram KFLM. Therefore the parallelogram KFLM has been described equal to the given rectilineal figure ABCD, having the angle FKM equal to the given angle E. Which was to be done.
Cor. From this it is manifest how to a given straight line to apply a parallelogram, which shall have an angle equal to a given rectilineal angle, and shall be equal to a given rectilineal figure, viz. by applying* to the given straight line a parallelogram equal to the first triangle ABD, and having an angle equal to the given angle.
PROP. XLVI. PROB.
To describe a square upon a given straight line. Let AB be the given straight line; it is required to describe a square upon AB.
From the point À draw* AC at right angles to AB; . 11. 1. and make* AD equal to AB: through the point D draw * 3. 1. DE parallel* to AB, and through B draw BE parallel 31. 1. to AD; therefore ADEB is at parallelogram: whence + Def. 34. AB is equal* to DE, and AD to BE: but BA is equalt to AD; therefore the four straight lines BA, AD, DE, EB, are equalt to one another, and the parallel- D ogram ADEB is equilateral : likewise all its angles are right angles; for, since the straight line AD meets the parallels AB, DE, the angles BAD, ADE are
+ 1 Ax.
* 29. 1.
+ Constr. + 3 Ax.
* 34. 1.
* 46. 1. * 31. 1.
equal* to two right angles: but BAD is at right angle, therefore also ADE is at right angle: but the opposite
angles of parallelograms* are equal; therefore each of +1 Ax.
the opposite angles ABE, BED is at rigbt angle; wherefore the figure ADEB is rectangular: and it has
been demonstrated that it is equilateral; it is therefore + 30 Def.
at square, and it is described upon the given straight line AB. Which was to be done.
Cor. Hence every parallelogram that has one right angle, has all its angles right angles.
PROP. XLVII. THEOR.
upon the side subtending the right angle, is equal to the
Let ABC be a right-angled triangle, having the right angle BAC: the square described upon the side BC shall be equal to the squares described upon BA, AC.
On BC describe* the square BDEC, and on BA, AC the squares GB, HC; and through A draw* AL
parallel to BD, or CE, and join AD, FC. Then, + Hyp. because the angle BAC is at right angle, and that the 30 Def. angle BAG is also a* right angle,
the two straight lines AC, AG upon
line. And because the angle DBC † 11 Ax.
is equalt to the angle FB, each of + 30 Def. them being a rightf angle, add to each the angle ABC;
therefore the whole angle DBA is equal* to the whole + 30 Def.
FBC: and because the two sides AB, BD, are equalt to the two FB, BC, each to each, and the angle DBA equal to the angle FBC; therefore the base AD is equal* to the base FC, and the triangle ABD to the triangle FBC: now the parallelogram BL is double* of the triangle ABD, because they are upon the same base BD, and between the same parallels BD, AL; and the square GB is double of the triangle FBC, because these also are upon the same base FB, and
* 14. 1.
* 2 Ax.
* 4. 1. * 41.1.
between the same parallels FB, GC: but the doubles of equals are equal* to one another; therefore the * 6 Ax. parallelogram BL is equal to the square GB. In the same manner, by joining AE, BK, it can be demonstrated, that the parallelogram CL is equal to the square HC: therefore the whole square BDEC is equalt to the two squares GB, HC: and the square + 2 Ax. BDEC is described upon the straight line BC, and the squares GB, HC upon BA, AC; therefore the square upon the side BC is equal to the squares upon the sides BA, AC. Therefore, in any right-angled triangle, &c.
Q. E. D.
PROP. XLVIII. THEOR.
If the square described upon one of the sides of a triangle,
be equal to the squares described upon the other two sides of it; the angle contained by these two sides is a right angle.
Let the square described upon BC, one of the sides of the triangle ABC, be equal to the squares upon the other sides BA, AC: the angle BAC shall be a right angle.
From the point A draw* AD at right angles to AC, 1. 1. and maket AD equal to BA, and join DC. Then, + 3. 1. because DA is equal to AB, the square of DA is equal to the square of AB: to each of these add the square of AC; therefore the squares of DA, AC are equalt to + 2 Ax. the squares of BA, AC: but the square of DC is equal* to the squares of DA, AC, because DAC is at right angle; and the
+ Constr. square of BC, by hypothesis, is equal to the squares of BA, AC; therefore the square of DC is equalt to the square of
+1 Ax. BC; and therefore also the side DC is equal to the side BC. And because the side DA is equalt to AB, and AC common to the two triangles + Constr. DAC, BAC, the two DA, AC are equal to the two BA, AC each to each; and the base DC has been proved equal to the base BC; therefore the angle DAC is equal* to the angle BAC: but DAC is at right angle; therefore also BAC is at right angle. There- Constr.
+1 Ax fore, if the square, &c. Q. E. D.
* 47. 1.
Every right-angled parallelogram, or rectangle, is said to
be contained by any two of the straight lines which contain one of the right angles.
In every parallelogram, any of the parallelograms about
a diameter, together with the two
K . FC, is the
which is more . briefly expressed by the letters
AGK, or EHC, which are at the opposite angles
PROP. I. THEOR.
any number of parts; the rectangle contained by the two
Let A and BC be two straight lines; and let BC be divided into any parts in the points D, E: the rectangle
K L H
* 11. 1.
3. 1. • 31. 1.
contained by the straight lines A, BC shall be equal to the rectangle con
DEC tained by A, BD, together with that contained by A, DE, and that contained by A, EC. From the point B draw* BF at
A right angles to BC, and make BG equal * to A; and through G draw* GH parallel to BC; and through D, E, C, draw* DK, EL, CH pa- . 31. 1. rallel to BG. Then the rectangle BH is equal to the rectangles BK, DL, EH: but BH is contained by A, BC, for it is contained by GB, BC, and GB is equal + to A; and BK is contained by A, BD, for it is + Constr. contained by GB, BD, of which GB is equal to A; and DL is contained by A, DE, because DK, that is * BG, * 34. 1. is equal to A; and in like manner the rectangle EH is contained by A, EC: therefore the rectangle contained by A, BC, is equal to the several rectangles contained by A, BD, and by A, DE, and by A, EC. Wherefore, if there be two straight lines, &c. Q. E. D.
PROP. II. THEOR.
If a straight line be divided into any two parts, the rect
angles contained by the whole and each of the parts,
* 46.1. # 31. 1.
Let the straight line AB be divided into any two parts in the point C: the rectangle contained by AB, BC, together with the rectangle | AB, AC shall be equal to the square of AB. Upon A B describe* the square ADEB,
C B and through C draw* CF, parallel to AD or BE. Then AE is equal to the rectangles AF, CE: but AE is the square of AB; and AF is the rectangle contained
F E by BA, AC; for it is contained by DA, AC, of which AD is equal+ to AB: and CE is contained + 30 Def. by AB, BC, for BE is equal to AB: therefore the rectangle contained by AB, AC, together with the rect
# N. B.--To avoid repeating the word contained too frequently, the rectangle contained by two straight lines AB, AC is sometimes simply called the rectangle AB, AC.