...X2 + Y2 = (X + Y + i)1 X2 + Y«= (X — Y -fi)2 (7 Marks). 2. Divide a straight line into two parts so that the rectangle contained by the whole and one part shall be equal to the square on the other part. (io Marks). 3. A battalion of soldiers when formed into a solid square present 16...

...to the Pythagoreans. The construction of this triangle depends upon u. n, or the problem of dividing a straight line so that the rectangle contained by the whole and one of the parts is equal to the square on the other part. This problem of course appears again in Eucl....

...of the triangle, though the latter brings out the result more easily. PROPOSITION n. To cut a given straight line so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment. Let AB be the given straight line...

...Proposition II. 1 1 of the Elements asks to divide a line into mean and extreme ratio: To cut a given straight line so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment.49 The basis of the construction provided...

...given figure q." Similarly, x2—a(a—x) of the third type is Euclid's proposition "To cut a given straight line so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment."* And Euclid finds ж— i/a2 + (a/2)*...

...- b) + b2 = a2 which is a variation on (a + b)(a - b) = a2 -- b2. Theorem 7 (II, 11) To cut a given straight line so that the rectangle contained by the whole and one of the parts is equal to the square of the other. Let the given line be AB. The problem is to find...