The elements of Euclid, explained and demonstrated in a new and most easie method with the uses of each proposition in all the parts of the mathematicks / by Claude Francois Milliet D'Chales, a Jesuit ; done out of French, corrected and augmented, and illustrated with nine copper plates, and the effigies of Euclid, by Reeve Williams ...

PROPOSITION XIII. THEOREM.

IN any Triangle whatever, the Square of the side opposite to an acute Angle, together with two Rectangles comprehend∣ed under the side on which the Perpendi∣cular falleth, and under the Line which is betwixt the Perpendicular and that An∣gle, is equal to the Square of the other sides.

Let the proposed Triangle be ABC, which hath the Angle C, acute; and if one draw AD perpendicular to BC, the Square of the side AB, which is opposite to the acute Angle C, toge∣ther with two Rectangles comprehend∣ed under BC, DC, shall be equal to the Squares of AC, BC.

Demonstration. The Line BC is di∣vided in D; whence (by the 7^th.) the Square of BC, DC, are equal to two Rectangles under BC, DC, and to

the Square of BD: add to both the Square of AD; the Square of BD, DC, AD, shall be equal to two Re∣ctangles under BC, DC; and to the Squares of BD, AD, in the place of the Squares of CD, AD, put the Square of AC, which is equal to them (by the 47^th. of the 1^st.) and instead of the Squares of BD, AB, substitute the Square of AB, which is equal to them, the Squares of BC, AC, shall be equal to the Square of AB, and to two Rect∣angles comprehended under BC, DC.

USE.

THese Propositions are very necessary in Trigonometry: I make use there∣of in the eighth Proposition of the third Book, to prove, That in a Triangle there is the same Reason between the whole Sine and the Sine of an Angle, as are between the Rectangle of the sides comprehending that Angle, and the double Area of the Triangle. I make use thereof in several other Propositions, as in the seventh.