PROP. XXXIX. THEOR. 47. 1 Eu.

In any right-angled triangle, the square which

is described upon the hypothenuse, or side
subtending the right angle, is equal to the
sum of the squares described upon the sides
which contain the right angle.
Let ABC be a rt. angled , having the rt.
BAC. Then BC* = BA? + AC?.

D LE

PROP. XL. THEOR.
If one side of a triangle be produced, and the

exterior angle, also one of the interior and
opposite angles, be each of them bisected ;
the remaining interior and opposite angle
will be double the angle made by the bisect-
ing lines.

Let CBA be a A. Prod. BC to D, and let the ext. L ACD be bisected by the str. line CE, and the int. and opp. L ABC by the str. line BE. Then will Z BAC = 2L BEC.

DEFINITIONS.

Every right-angled parallelogram, or rect

angle, is said to be contained by any two of the straight lines which contain one of the right angles.

II. In every parallelogram, any of the parallelo

grams about a diameter, together with the two complements, is called a Gnomon. « Thus the parallelogram HG, together with the complements AF, FC is the Gnomon; which is more briefly expressed by the letters AGK or EHC, which are at the opposite angles of the parallelograms which make the gnomon.'

A

Gc N.B.-The rectangle contained by any two straight lines AB, AD, is generally expressed by the words “rectangle of AB and AD;" or by “AB. AD.”

PROP. XLI. THEOR. 1. 2 Eu. If there be two straight lines, one of which is

divided into any number of parts; the rect