tained by the two parts, which are the greatest and the least, is less than the rectangle contained by the other two parts; the squares of the two former parts, together, are greater than the squares oft he two latter, taken together; and the difference between the squares of the former and the squares of the latter, is the double of the difference between the two rectangles. (m.) In any isosceles triangle, if a straight line be drawn from the vertex to any point in the base, the square upon this line, together with the rectangle contained by the segments of the base, is equal to the square upon either of the equal sides. PROP. VI. (IV.) The rectangle contained by the aggregate and the difference of two unequal straight lines is equal to the difference of their squares. COR. If there be three straight lines, the difference between the first and second of which is equal to the difference between the second and third, the rectangle contained by the first and third, is less than the square of the second, by the square of the common difference between the lines. PROP. VII. (v.) The square of the excess of the greater of two given straight lines above the less, is less than the squares of the two lines, by twice the rectangle contained by them. (VI.) The squares of any two unequal straight lines are, together, greater than twice the rectangle contained by those lines. PROP. VIII. (VII.) If a straight line be divided into five equal parts, the square of the whole line is equal to the square of the straight line, which is made up of four of those parts, together with the square of the straight line which is made up of three of those parts. (VIII.) Upon a given straight line, as an hypotenuse, to describe a right-angled triangle, such that the hypotenuse, together with the less of the two remaining sides, shall be the double of the greater of those sides. PROP. X.. (IX.) of the two sides are, In any triangle, the squares together, the double of the squares of half the base, and of the straight line joining, its bisection and the opposite angle. (x.) Hence, the squares of the sides of any parallelogram are, together, equal to the squares of its diameters taken together.* (XI.) If either diameter of a parallelogram be equal to one of the sides about the opposite angle of the figure, its square shall be less than the square of the other diameter, by twice the square of the other side about that opposite angle. * This proposition may also be readily deduced from E. 13. 2. PROP. XII. (XII.) If two sides of a trapezium be parallel to each other, the squares of its diagonals are, together, equal to the aggregate of the squares of its two sides, which are not parallel, and of twice the rectangle of its parallel sides. PROP. XIII. (XIII.) The square of the base of an isosceles triangle is the double of the rectangle contained by either side, and by the straight line intercepted between the perpendicular, let fall upon it from the opposite angle, and the extremity of the base. (XIV.) If from any point, in the circumference of the greater of two given concentric circles, two straight lines be drawn to the extremities of any diameter of the less, their squares shall be, together, the double of the squares of the two semi-diameters of the two given circles. DEDUCTIONS FROM THE FIRST SIX BOOKS OF Euclid's Elements. BOOK III. PROP. I. (1.) Ir two circles cut each other, the straight line joining their two points of intersection is bisected, at right angles, by the straight line joining their centers. COR. Hence, if a trapezium have two of its adjacent sides equal to one another, and also its two remaining sides equal to one another, its diameters bisect each other at right angles. |