3. On BE construct a square BEFG equal to the rectangle ABCD (Pr. 167); then BEFG shall have an area of two square inches. Problem 169. To construct a square equal in area to any given triangle, ABC. 1. Make the rectangle DBCE equal to the triangle ABC (Pr. 159). A E H 2. Find CF a mean proportional to the lines BC, CE Problem 170. To construct a square having an area one-third greater than that of a given square ABCD. 1. Divide BC into three equal parts (Pr. 15), and from the points of division draw lines parallel to BA or CD (Pr. 9). 2. Produce BC beyond C, and AD beyond D, and make CE and DF each equal to one-third of BC. 3. Join EF; and find a mean proportional, EG (Pr. 140), to two adjacent sides of the rectangle BEFA. 4. Complete the required square of which EG is a side (Pr. 34). Then the square EGHK is one-third greater than the given square ABCD. Problem 171. To construct a square having an area one-third less than that of a given square ABCD. 1. Divide BC into three equal parts (Pr. 15), and from the points of division draw lines parallel to BA or CD (Pr. 9). 2. Find a mean proportional AE (Pr. 140) to the two 3. Complete the required square, of which AE is one side Problem 172. To inscribe within a given square, ABCD, another square having its angles in the sides of the first, and being proportional in area, as 3 2 1. Divide BC into three equal parts (Pr. 15), make DE equal to C 1 and join 1 E. 2. Find a mean proportional DF (Pr. 140), to the two sides CD, DE, of the rectangle 1 CDE, which will be equal to one side of the required square. 3. Make DG equal to DF, and join FG, which is equal to one diagonal of the required square. 4. Bisect FG in H (Pr. 1), and draw the diagonals AC, BD, cutting in K. 5. From K, with radius HF or HG, describe a circle cutting the sides of the given square in two points each. Join the alternate points L, M, N, O, and the required square will be inscribed within the given square ABCD. Problem 173. To construct a triangle equal in area to a given trapezium, ABCD. 1. Draw a diagonal DB, and produce AB indefinitely to E. C E 2. Through C, draw a line CE parallel to DB (Pr. 9), meeting AB produced in E. 3. Join DE; then ADE will be a triangle equal in area to the given trapezium ABCD. Problem 174. To construct a square equal in area to a given trapezium ABCD. 1. Construct a triangle BAE equal to the given trapezium (Pr. 173). 2. Construct a rectangle BFGE on BE equal to the triangle BAE, by bisecting the altitude AH (Pr. 159), producing the line of bisection FG, making it equal to BE, and joining BF and EG. 3. Construct a square equal in area to the rectangle BFGE, by finding a mean proportional EK (Pr. 140) to the two sides BE, EG. 4. Complete the required square EKLM, of which EK is one side (Pr. 34). Then the square EKLM is equal in area to the given trapezium ABCD. Problem 175. To construct a square equal in area to any number of squares, of which A, B, C, D, &c., are the given sides. A B C K H 1. Place A and B at right angles to each other, EF being equal to A, and FG to B, and join EG; then the square constructed on the hypotenuse EG is equal to the squares on A, B (Euc. I. 47). |