If two ratios are equal, their duplicate ratios are equal; and conversely, if the duplicate ratios of two ratios are equal, the ratios themselves are equal.

Let A B C : D;

then shall the duplicate ratio of A to B be equal to that of

C to D.

Let X be a third proportional to A and B, and Y a third proportional to C and D,

so that A B :: B: X, and C: D::

.., ex æquali,

then because A B C D,

.. BX :: D: Y;

AX C: Y.

: Y;

But A: X and C : Y are respectively the duplicate ratios of

A B and C : D,

.. the duplicate ratio of A : B that of C : D.

Def. 13.

Conversely, let the duplicate ratio of A: B = that of C : D; then shall A: B:: C: D.

Let P be such that A: B :: C: P,

that is, P is the mean proportional between C and Y.

.'. P = D,

.. A B C : D.

BOOK VI.

DEFINITIONS.

1. Two rectilineal figures are said to be equiangular when the angles of the first, taken in order, are equal respectively to those of the second, taken in order. Each angle of the first figure is said to correspond to the angle to which it is equal in the second figure, and sides adjacent to corresponding angles are called corresponding sides.

2. Rectilineal figures are said to be similar when they are equiangular and have the sides about the equal angles proportionals, the corresponding sides being homologous. [See Def. 5, page 288.]

Thus the two quadrilaterals ABCD,
angles at A, B, C, D are respec-
tively equal to those at E, F, G, H,
and if the following proportions
hold

AB: BC:: EF: FG,
BC CD: FG : GH,
CD: DA: GH: HE,

DA AB: HE : EF.

A

EFGH are similar if the

B

E

H G

3. Two figures are said to have their sides about two of their angles reciprocally proportional when a side of the first is to a side of the second as the remaining side of the second is to the remaining side of the first.

4. A straight line is said to be divided in extreme and mean ratio when the whole is to the greater segment as the greater segment is to the less.

5. Two similar rectilineal figures are said to be similarly situated with respect to two of their sides when these sides are homologous.

PROPOSITION 1. THEOREM.

The areas of triangles of the same altitude are to one another as their bases.

H

B

K

M

Let ABC, ACD be two triangles of the same altitude, namely the perpendicular from A to BD:

then shall the A ABC: the ▲ ACD :: BC: CD.

Produce BD both ways,

and from CB produced cut off any number of parts BG, GH, each equal to BC;

and from CD produced cut off any number of parts DK, KL, LM each equal to CD.

Then the

Join AH, AG, AK, AL, AM.

ABC, ABG, AGH are equal in area, for they are of the same altitude and stand on the equal bases CB, BG, GH,

1. 38.

.. the AHC is the same multiple of the ▲ ABC that HC is of BC;

Similarly the ACM is the same multiple of ACD that CM is of CD.

And if HCCM,

the AAHC the ACM;

=

I. 38.

ACM;

1. 38, Cor.

and if HC is greater than CM,

the AHC is greater than the

and if HC is less than CM,
AHC is less than the ACM.

the I. 38, Cor. Now since there are four magnitudes, namely, the ▲ ABC, ACD, and the bases BC, CD; and of the antecedents, any equimultiples have been taken, namely, the AHC

and the base HC; and of the consequents, any equimultiples have been taken, namely the ACM and the base CM; and since it has been shewn that the AHC is greater than, equal to, or less than the ▲ ACM, according as HC is greater than, equal to, or less than CM;

.. the four original magnitudes are proportionals, v. Def. 4. that is,

the ABC the AACD :: the base BC: the base CD. Q.E.D.

:

COROLLARY.

The areas of parallelograms of the same

altitude are to one another as their bases.

Let EC, CF be parms of the same altitude;

then shall the parm EC: the parm CF :: BC: CD.

Join BA, AD.

Then the A ABC: the AACD :: BC: CD; Proved. but the parm EC is double of the

and the par

CF is double of the

.. the par

ABC,

ACD;

NOTE. Two straight lines are cut proportionally when the segments of one line are in the same ratio as the corresponding segments of the other. [See definition, page 131.]

Thus AB and CD are cut proportionally at X and Y, if

AX: XB: CY: YD.

And the same definition applies equally whether X and Y divide AB, CD internally as in Fig. 1 or externally as in Fig. 2.

PROPOSITION 2. THEOREM.

If a straight line be drawn parallel to one side of a triangle, it shall cut the other sides, or those sides produced, proportionally:

Conversely, if the sides or the sides produced be cut proportionally, the straight line which joins the points of section, shall be parallel to the remaining side of the triangle.

A

A

X

B

B

Let XY be drawn par1 to BC, one of the sides of the A ABC:

.. the BXY: the AAXY:: the ACXY: the AAXY. v. 4.

But the ABXY: the AAXY :: BX: XA,

and the ACXY: the AAXY:: CY: YA,

.. BX XA :: CY: YA.

VI. 1.

v. 1.

Conversely, let BX: XA :: CY: YA, and let XY be joined: then shall XY be par1 to BC.

As before, join BY, CX.

By hypothesis BX : XA :: CY: YA;

but BX XA :: the ABXY: the AAXY, vi. 1.

and CY YA :: the ACXY: the AAXY;

:

.. the ABXY: the AAXY:: the A CXY: the AAXY. v.1. .. the BXY = the ACXY;

V. 6.

and they are triangles on the same base and on the same side of it.