...figures HP, CK,AG, GE, are equal to the squares of AC, CB, and to twice the rectangle AC, CB: but HF, CK, AG, GE, make up the whole figure ADEB, which is the square of AB: therefore 16. The square of AB is equal to the squares of AC, CB, and twice the rectangle AC, CB. Wherefore,...

...С k, ag, ge are equal to the squares of aС, Сb, and to twice the rectangle aС, Сb. But hf, Сk, ag, ge make up the whole figure adeb, which is the square of a b. Therefore the square of ab is equal to the squares of a С, С b, and twice the rectangle a С,...

...the rectangle contained by the parts. Let the straight line AB be divided into any two parts in C ; the square of AB is equal to the squares of AC, CB, and to twice the rectangle contained by AC, CB. Upon AB describe (46. i.) the square ADEB, and join BD,...

...with twice the rectangle AС.СB. But the figureа HF, СK, AG and GE make up the whole figure AE, which is the square of AB. Therefore the square of AB is equal to the squares of AС and СB, and twice the rectangle AС.СB. Wherefore, if a straight line be divided, &c. QED Сов....

...rectangle contained by the parts. Let the straight line AB (Fig. 39) be divided into any two parts in C. The square of AB is equal to the squares of AC, CB and twice the rectangle contained by AC, C B. Let AE be the square on AB, BD its diameter, CGF parallel to AD or BE, cutting...

...whole figure ADEB, which is the square on AB ; therefore the square on AB is equal to the squares on AC, CB, and twice the rectangle AC, CB. Wherefore, if a straight line be divided, &c. QED COB. From the demonstration, it is manifest, that the parallelograms about the...

...therefore the square on AB is equal to the squares on AC and CB, and twice the rectangle AC • CB. COR. — From the demonstration, it is manifest that the parallelograms about the diagonal of a square are likewise squares. PROPOSITION V. THEOREM. -;I| If a straight line be divided...

...because the angle at D is a right angle, the angle ACB is greater than a right angle ; (i. 16) and therefore the square of AB is equal to the squares of AC, CB, and twice ti,e rectangle BC, CD; (Ii. 12) to each of these equals add the square of BC, then the squares of AB,...

...AC and CB. to the squares on AC and CB, together with twice the rectangle AC . CB. 21. But HF, CK, AG, GE, make up the whole figure ADEB, which is the square on AB. 22. Therefore the square on AB is equal to the squares on AC and CB, and twice the rectangle...

...C draw CGF parallel to AD or BE (I. 31), and through G draw HGK parallel to AB or DE. K Cor.—From the demonstration, it is manifest, that the parallelograms...about the diameter of a square, are likewise squares. PEOP. V. (THEonEM.)—Tf a straight line (AB) be divided into two equal parts (at C), and also into...