therefore the rectangle BE, EF, together with the square of EG, is equal to the square of GH: But the squares of HE and EG are equal (37. 1.) to the square of GH: Therefore also the rectangle BE.EF, together with the square of EG, is equal to the squares of HE and EG. Take away the square of EG, which is common to both, and the remaining rectangle BE.EF is equal to the square of EH: But BD is the rectangle contained by BE and EF, because EF is equal to ED; therefore BD is equal to the square of EH ; and BD is also equal to the rectilineal figure A; therefore the rectilineal figure A is equal to the square of EH: Wherefore a square has been made equal to the given rectilineal figure A, viz. the square described upon EH.

Note. This operation is called squaring the rectilineal figure, or finding the quadrature of it.

PROP. II. PROB.

To construct a rectangle that shall be equivalent to a given square, and the difference of whose adjacent sides shall be equal to a given line.

Suppose C equal to the given square, and AB the difference of the sides.

Upon the given line AB as

:

a diameter, describe a semicircle at the extremity of the diameter draw the tangent AD equal to the side of the square C; through the point D, and the centre O, draw the

secant DF; then will DE and DF be the adjacent sides of the rectangle required.

First, the difference of their sides is equal to the diameter EF or AB; secondly, the rectangle DE.DF is equal to AD2 (32. 3.); hence that rectangle is equivalent to the given square C.

PROP. III. PROB.

To construct a rectangle equivalent to a given square, and having the sum of its adjacent sides equal to a given line.

Let C be the given square, and AB equal to the sum of the sides of the required triangle.

Upon AB as a diameter, describe a semicircle; draw the line DE parallel to the diameter, at a distance AD from it, equal to the side of the given square C; from the point E, where the parallel cuts the circumference, draw EF perpendicular to the diameter; AF and FB will be the sides of the rectangle required.

D

For their sum is equal to AB; and their rectangle AF.FB is equal to the square EF, or to the square AD; hence that rectangle is equivalent to the given square C.

SCHOLIUM.

To render the problem possible, the distance AD must not exceed the radius; that is, the side of the square C must not exceed the half of the line AB.

PROP. IV. PROB.

To construct a square that shall be to a given square as a given line to a given line.

Upon the indefinite straight line GH take GK=E, and KH=F; describe on GH a semicircle, and draw the perpendicular KL. Through the points G, H, draw the straight lines LM, LN, making the former equal AB, the side of the given square, and through the point M, draw MN parallel to GH, then will LN be the side of the square sought.

E

F

D

L

G K

H

A

B

M

N

For, since MN is parallel to GH, LM: LN:: LG : LH; consequently, LM2: LN2 :: LG2: LH2 (22. 6.); but, since the triangle LGH is right angled, we have LG2 : LH2:: GK: KH; hence LM2: LN2 :: GK: KH; but, by construction GK=E, and KH=F, also LM =AB; therefore, the square described on AB is to that described on LN, as the line E is to the line F.

PROP. V. PROB.

To divide a triangle into two parts by a line from the vertex of one of its angles, so that the parts may be to each other as a straight line M to another straight

line N.

Divide BC into parts BD, DC proportional to M, N; draw the line AD, and the triangle ABC will be divided as required.

For, since the triangles of the same altitude are to each other as their bases,

we have ABD: ADC:: BD: DC :: B D M: N.

SCHOLIUM.

A

A triangle may evidently be divided into any number of parts proportional to given lines, by dividing the base in the same proportion.

PROP. VI. PROB.

To divide a triangle into two parts by a line drawn parallel to one of its sides, so that these parts may be to each other as two straight lines M, N.

As M+N: N, so make AB2 to AD2 (Prob. 4.); Draw DE parallel to BC, and the triangle is divided as required.

For the triangles ABC, ADE being similar, ABC :ADE:: AB2: AD2; but M+N: :N:: AB2: AD2; therefore ABC ADE M+N: N; consequently BDECADE:: M: N.

B

E

PROP. VII. PROB.

To divide a triangle into two parts, by a line drawn from a given point in one of its sides, so that the parts may be to each other as two given lines M, N.

Let ABC be the given triangle, and P the given point; draw PC, and divide AB in D, so that AD is to DB as M is to N; draw DE parallel to PC, join PE, and the triangle will be divided by the line PE into the proposed parts.

For join DC; then because PC, DE are parallel, the triangles PDE, CDE are equal; to each add the triangle DEB, then PEB= DCB; and consequently, by taking each from the triangle ABC, there results the quadrilateral ACEP equivalent to the triangle ACD.

B

D

P

Now, ACD: DCB :: AD: DB:: M: N; consequently,
ACEP PEB:: M: N

SCHOLIUM.

The above operation suggests the method of dividing a triangle into any number of equal parts by lines drawn from a given point in one of its sides; for if AB be divided into equal parts, and lines be drawn from the points of equal division, parallel to PC, they will intersect BC, and AC; and from these several points of intersection if lines be drawn to P, they will divide the triangle into equal parts.

PROP. VIII. PROB.

To divide a triangle into three equivalent parts by lines drawn from the vertices of the angles to the same point within the triangle.

Make BD equal to a third part of BC, and draw DE parallel to BA, the side to which BD is adjacent. From F, the middle of DE, draw the straight lines FA, FB, FC, and they will divide the triangle as required.

For, draw DA; then since BD is one third of BC, the triangle ABD is one third of the triangle ABC; but ABD= ABF (31. 1.); therefore ABF is one third of ABC; also, since DF=FE, BDF = AFE; likewise CFD = CFE, consequently the whole triangle FBC is equal to the whole triangle FCA; and

F

E

FBA has been shown to be equal to a third part of the whole triangle ABC; consequently the triangles FBA, FBC, FCA, are each equal to a third part of ABC.

PROP. IX. PROB.

To divide a triangle into three equivalent parts, by lines drawn from a given point within it.

Divide BC into three equal parts in the points D, E, and draw PD, PE; draw also AF parallel to PD, and AG parallel to PE; then if the lines PF, PG, PA be drawn, the trian

gle ABC will be divided by them into three equivalent parts.

A

For, join AD, AE; then because AF, PD are parallel, the triangle AFP is equivalent to the triangle AFD; consequently, if to each of these there be added the triangle ABF, there will result the quadrilateral ABFP equivalent to the triangle ABD; but since BD is a third part of BC, the triangle ABD is a third part of the triangle ABC; consequently the quadrilateral ABEP is a third part of the triangle ABC. Again, because AG, PE are parallel, the triangle AGP is equivalent to the triangle ACE; and this triangle is one third of ABC; hence the quadrilateral ACGP is one third of the triangle ABC: consequently, the spaces ABFP, ACPG, PFG are each equal to a third part of the triangle ABC.

FD

E

C

PROP. X. PROB.

To divide a quadrilateral into two parts by a straight line drawn from the vertex of one of its angles, so that the parts may be to each other as a line M to another line N.

D

Draw CE perpendicular to AB, and construct a rectangle equivalent to the given quadrilateral, of which one side may be CE; let the other side be EF; and divide EF in G, so that M:N GF: EG; take BP equal to twice EG, and join PC, then the quadrilateral will be divided as required.

For, by construction, the triangle CPB is equivalent to the rectangle CE.EG; therefore the rectangle CE, GF is to the triangle CPB as GF is to EG. Now CE.GF is equivalent

P

BEG

F

to the quadrilateral DP, and GF is to EG as M is to N; therefore,

DP CPB:: M: N;

that is, the quadrilateral is divided, as required.

PROP. XI. PROB.

To divide a quadrilateral into two parts by a line parallel to one of its sides, so that these parts may be to each other as the line M is to the line N.

Produce AD, BC till they meet in E; draw the perpendicular EF and bisect it in G. Upon the side GF construct a rectangle equivalent to the triangle EDC, and let HB be equal to the other side of this rectangle. Divide AH in K, so that AK : KH :: M: N, and as AB is to KB, so make DA2 to Ea2; draw ab parallel to AB, and it will divide the quadrilateral into the required parts.

:

A

E

AC

D

G

K F H B

For since the triangles EAB, Eab are similar, we have the proportion EAB Eab:: EA2: Ea2; but by construction, EA2: Ea2 :: AB: KB; so that EAB: Eab::AB: KB:: AB.GF: KB.GF; and consequently, since by construction EAB=AB.GF, it follows that Eab=KB. GF, and therefore AK.GF-Ab, and since by construction AH.GF=AC, it follows that KH.GF=aC. Now AK.GF: KH.GF: AK: KH; but