The First Six Books of the Elements of Euclid

by Euclid; John Casey

	Or thus:	AB	= AC + CB.
	Squaring, we get	AB2 =	AC2 + 2AC.CB + CB2.

Cor. 1.—The parallelograms about the diagonal of a square are squares.

Cor. 2.—The square on a line is equal to four times the square on its half.

For let AB = 2AC, then AB² = 4AC².

This Cor. may be proved by the First Book thus: Erect CD at right angles to AB, and make CD = AC or CB. Join AD, DB.

	Then	AD2 = AC2	+ CD2 = 2AC2
	In like manner,	DB2	= 2CB2;
	therefore	AD2 + DB2	= 2AC2 + 2CB2 = 4AC2.

But since the angle ADB is right, AD² + DB² = AB²;

therefore

AB2

= 4AC2.

Cor. 3.—If a line be divided into any number of parts, the square on the whole is equal to the sum of the squares on all the parts, together with twice the sum of the rectangles contained by the several distinct pairs of parts.