Prop. 67.

That is, DH: AG :: EF: BC;
Or DH: EF:: AG: BC.

Therefore, equiangular As, &c.

Triangles having equal bases are to each other as their altitudes.

Let ACB, DFE be 2 s, having base AB
Draw CHLAB, and FILDE,
ACB: DFE :: CH: FI.

base DE.

then shall

And APB: MAQB :: PB: QB,
A ACB: ADFE:: CH: FI.

Prop. 71.

Wherefore triangles, &c.

PROP. LXXXIII. THEOR. 19. 6 Eu. Equiangular triangles are to each other as the squares of their corresponding sides.

Let As ABC, DEF have the s A, B, C, s D, E, F respectively; then shall AABC: A DEF:: BC2: EF2.

=

On BC and EF draw the squares BK, EM,

and the diams. CI, FL,

Also draw AGL BC and DHL EF.

Prop. 38.

Since As on = bases are as their altitudes, Prop.82.

ДАВС AG

AG

Hyp.

Prop. 81.

Ax. 1.

Ax. 7.

Prop. 67.

or AABC: ADEF :: BC2 : EF2.

Therefore, equiangular As, &c.

PROP. LXXXIV. THEOR.

Similar rectilinear figures are to each other as the squares of their like or corresponding sides.

Let ABCDE, FGHIK be any two similar figures, having the s A, B, C, &c., of the one figure respectively equal to the s F, G, H, &c., of the other, and the sides about the equals proportional; also let AB and FG be corresponding sides; then shall fig. ABCDE : fig. FGHIK :: AB2 : FG2.

Draw BE. BD, GK, GI dividing the figs. into an = number of As, by lines from the equal Ls B and G.

Then

Also

Again...

<C= <H,

BC: CD :: GH : HI,

Hyp.

Prop. 74.

Hyp.

JAS BCD, GHI are equiangular. Prop. 74.

4CDB

HIG.

ZAEB ZFKG,

.. rem. BED =rem. GKI.

and

.. rem.

[CDE HIK,

=

BDE

HIG,
=rem. GIK;

.. As BDE, GIK are equiangular.

each of the one fig. is equiangular and similar to each corresponding of the other

fig.

AB2

ΔΑΒΕ ABCD

Ax. 1.

FG2

ABDE AFGK AGHI AGIK

Prop. 68... AB2 : FG2 : : ▲ ABE + ▲ BCD + A BDE: FGK + ◇GHI + AGIK; or AB2: FG2:: whole fig. ABCDE: whole fig. FGHIK.

In like manner it may be proved that similar rectilinear figures of any number of sides, are as the squares of their like or corresponding sides.

Wherefore similar rectilinear figures, &c.

The circumferences of equilateral polygons, which have the same number of sides, have the same ratio as the radii of their circumscribing circles.

Let ABCDEF, GHKILM be 2 equilat. polygons having anno. sides inscrib. in Os whose centres are Q, R; join AQ, BQ, &c. and GR, HR, &c. : then shall Oce of polygon ABC, &c. ::" of polygon GHI &c. :: AQ : GR.