PROP. IX. THEOR. If a straight line be divided equally and unequally, the squares of the unequal parts are together double of the squares of half the line, and of the part intermediate between the points of section. If a straight line be bisected, and produced to any point, the squares of the whole produced line, and of the produced part, are together double of the squares of the half line, and of the line made up of the half and the produced part. PROP. XI. PROB. To divide a given straight line in such a manner that the rectangle contained by the whole line and one of the parts may be equal to the square of the other part. In an obtuse-angled triangle, the square of the side subtending the obtuse angle exceeds the sum of the squares of the sides containing the obtuse angle, by twice the rectangle contained by either of those sides and the produced part of it intercepted between the perpendicular let fall on it from the opposite angle and the vertex of the obtuse angle. C PROP. XIII. THEOR. In any triangle, the square of a side subtending an acute angle is less than the sum of the squares of the sides containing that angle, by twice the rectangle contained by either of those sides and the part of it intercepted between the perpendicular let fall on it from the opposite angle and the vertex of the acute angle. A A PROP. XIV. PROB. To draw a line, of which the square shall be equal to a given rectilinear figure. |