PROP. LXXX. THEOR.
The rectangle contained by the diagonals of a

quadrilateral figure inscribed in a circle,
is equal to both the rectangles together, con-
tained by its opposite sides.

Let ABCD be any quadrilateral fig. inscrib. in a 0 ; draw the diagonals AC, BD: then shall AC. BD = AB.CD + AD.BC.

PROP. LXXXI. THEOR. Equiangular triangles have their bases in the same ratio as their altitudes, or perpendiculars upon the bases from the opposite and equal angles.

Let the As ABC, DEF have the Ls A, B, C of the one A, respectively equal to the Ls D, E, F of the other; draw AG I BC and DH I EF; then shall DH : AG :: EF: BC; or DH : EF :: AG : BC.

PROP. LXXXII. THEOR. Triangles having equal bases are to each other

as their altitudes. Let ACB, DFE be 2 As, having base AB = base DE. Draw CH | AB, and FI IDF, then shall AACB : ADFE :: CH : FI.

Р

Join SAP

make JPB = CH,

JAP

AQ
J ACB,-having the same Prop. 34.
1 bases and altitudes. Cor.

ADFE, - having equal Prop. 35.
KAAQB=1 bases and altitudes.
And AAPB : AAQB :: PB : QB, Prop. 71.

.. A ACB: ADFE:: CH: FI.
Wherefore triangles, &c.

Cor. 1.

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 Ls A, B, C, = Ls D, E, F respectively; then shall A ABC : A DEF :: BCP : EF.

On BC and EF draw the squares BK, EM, and the diams. CI, FL;

Prop. 38.
Also draw AG I BC and DH I EF.
Since As on = bases are as their altitudes, Prop. 82.

SAABC AG
JOBIC BIBC:
ADEF I
ZELF = EL = EF

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 Zs A, B, C, &c., of the one figure respectively equal to the Ls F, G, H, &c., of the other, and the sides about the equal Zs proportional; also let AB and FG be corresponding sides; then shall fig. ABCDE : fig. FGHIK :: AB? : FG%.