1. State and prove geometrically that (a+b+c)2= a2+b2+c2+2ab+2ac+2bc. 2. Of all rectangles of the same perimeter the square has the greatest area. 3. Prove Prop. 8 by means of Props. 4 and 7. 4. Show how to divide a given straight line into two parts so that the difference of the squares on the parts may be equal to a given square. 5. If a line AB be divided in C so that the rectangle AB, BC is equal to the square on AC, prove that the sum of the squares on AB and BC is equal to three times the square on AC. 6. Three times the sum of the squares of the sides of a triangle is equal to four times the sum of the squares of its three medians. LIV. 1. In any quadrilateral figure the squares on the diagonals are together equal to twice the sum of the squares on the straight lines joining the middle points of adjacent sides. 2. If a line AB is divided equally at C and unequally at D, prove that the difference of the squares on AD, DB is equal to twice the rectangle AB, CD. 3. The line AB is divided into any two parts at C, and produced to D, making BD equal to BC. Show that four rectangles, each equal to the rectangle AB, BC can be cut from the square on AD, one rectangle being taken at each of its corners, so as to leave a square equal to the square of AC. 4. Show how to produce a given straight line so that the rectangle contained by the whole line thus produced and the part produced may be equal to the square of the original line. 5. If a line AB be divided at C so that the rectangle AB, BC is equal to the square of AC, prove that (AC+BC)2=5AC2. 6. The sum of the squares of the four sides of a quadrilateral figure is equal to the sum of the squares of its diagonals plus four times the square of the line joining the middle points of the diagonals. |