PROPOSITION VIII.

Similar polygons are to one another as the squares on their corresponding sides.

Let ABCDF, GHKMT be similar polygons.

Then shall they bear to one another the same ratio as the squares on their corresponding sides AB, GH.

Divide the polygons into triangles by straight lines drawn from B and H.

Then... FA: AB as TG: GH, and ▲ FAB = L TGH; ... as FAB, TGH are similar;

(VI. 4) and are to each other as squares on FB, TH. (vI. 7) DFA LMTG and ▲ BFA = L HTG ;

Again, .

=

.. L DFB = L MTH;

also DF: FA as MT: TG;

and FA FB as TG: TH;

:. DF: FB as MT: TH;

(V. 12)

.. As DFB, MTH are similar;

(VI. 4)

and are to each other as squares on FB, TH, (v1. 7) and are .. in the same ratio as as FAB, TGH. So each of the as in the polygon ABCDF: the corresponding ▲ in the polygon GHKMT in the same ratio.

.. all the as in the polygon ABCDF bear to all the As in polygon GHKMT the same ratio. (v. II) That is, the area of polygon ABCDF: area of polygon GHKMT as ▲ FAB : ▲ TGH,

(VI. 7)

and ... as square on AB: square on GH. COR. Each of the sides of the one polygon is to the corresponding side of the other as AB: GH

.. also the perimeter of the one is to the perimeter of the other in the same ratio.

(v. II)

PROPOSITION IX.

If four straight lines are proportional, the squares on these

lines are proportional.

A

K

B

H

Let AB, AH, AC, AK be cut off from the arms of

any A = the given st. lines and ... proportional.

Then shall the squares on AB, AH, AC, AK be proportional also.

Join BC, HK.

Then BC is | to HK,

and .. as ABC, AHK are similar;

(VI. I)

(1. 21)

.. ▲ ABC: ▲ AHK as square on AB : square on AH,

and also

(VI. 7)

as square on AC: square on AK;

(VI. 7)

... sq. on AB : sq. on AH as sq. on AC: sq. on AK. (v. 3)

COR. Hence also, if similar figures are described on AB and AH and similar figures on AC and AK,

since figure on AB : similar figure on AH as square on AB square on AH,

and figure on AC : similar figure on AK as square on AC: square on AK, it follows that

fig. on AB sim. fig. on AH as fig. on AC: sim. fig. on AK.

(v. 3)

PROPOSITION X.

In right-angled triangles the rectilineal figure described on the hypothenuse is equal to the similar and similarly described figures on the other two sides.

Let P, Q, R be figures similar and similarly described on the sides of the right-angled ▲ ABC.

Then the figure P on the hypothenuse BC is equal to the figures and R together.

Then on BC, AC are the similar as ABC, DAC,
and also the similar figures P, Q;

.... AABC:

AABC: ▲ DAC as P : Q; (vi. 9 Cor.)

so also

▲ ABC:

P

as ▲ DBA: R;)

(v. 6)

.. ΔΑΒC :

P

as ▲ DAC together with a

DBA figures Q, R together,

but AABC is AS DAC, DBA together;

(v. II)

(v. 4)

PROBLEM F.

On a given straight line describe a figure similar and similarly situated to a given rectilineal figure.

Let CDEFG be the given rectilineal figure,

and XY the given straight line.

It is required to describe on XY a rectilineal figure similar and similarly situated to CDEFG.

Join DG, DF,

and make s YXR, XYR = 45 DCG, CDG, respectively, LS YRK, RYK = 4S DGF, GDF,

and LS YKH, KYH = L S DFE, FDE.

Then shall XYHKR be similar and similarly situated to CDEFG.

For the AS GCD, GDF &c. are respectively equiangular to the AS RXY, RYK &c.

.. CG: GD as XR : RY,
and GD: GF as RY: RK;

.. CG: GF as XR : RK,
and so on.

(VI. 4)

(VI. 4)

(v. 12)

the cor

Thus all the 4s in the one polygon are = responding 4s in the other, and the sides about the

correspondings are proportional.

..the polygons are similar.

PROBLEM G.

To describe a rectilineal figure equal to one and similar to another given rectilineal figure.

Let it be required to describe a rectilineal figure similar to ABCFG and equal to H.

Describe squares MS, PK equal to AC, and H.

(II. A)

To MN, PQ and AB take a fourth proportional XY. (vi. B) On XY describe a figure XZ similar to that on AB : (vi. F) this shall be the figure required.

For since MN: PQ as AB : XY;

... square MS: square PK as figure AC: figure XZ;

(VI. 9 Cor.)

but square MS = figure AC, .. square PK = figure XZ;

(v. 4)

.. figure XZ figure H, and it is similar to figure AC.