Exercises.
1. If the angle C of the △ ACB be equal to an angle of an equilateral △, AB2 = AC2 + BC2 − AC.BC.
2. The sum of the squares on the diagonals of a quadrilateral, together with four times the square on the line joining their middle points, is equal to the sum of the squares on its sides.
3. Find a point C in a given line AB produced, so that AC2 + BC2 = 2AC.BC.
PROP. XIV.—Problem.
To construct a square equal to a given rectilineal figure (X).
Sol.—Construct [I. xlv.] the rectangle AC equal to X. Then, if the adjacent sides AB, BC be equal, AC is a square, and the problem is solved; if not, produce AB to E, and make BE equal to BC; bisect AE in F; with F as centre and FE as radius, describe the semicircle AGE; produce CB to meet it in G. The square described on BG will be equal to X.
Dem.—Join FG. Then because AE is divided equally in F and unequally in B, the rectangle AB.BE, together with FB2 is equal to FE2 [v.], that is, to FG2; but FG2 is equal to FB2 + BG2 [I. xlvii.]. Therefore the rectangle AB.BE + FB2 is equal to FB2 + BG2. Reject FB2, which is common, and we have the rectangle AB.BE = BG2; but since BE is equal to BC, the rectangle AB.BE is equal to the figure AC. Therefore BG2 is equal to the figure AC, and therefore equal to the given rectilineal figure (X).
Cor.—The square on the perpendicular from any point in a semicircle on the diameter is equal to the rectangle contained by the segments of the diameter.
Exercises.