—Let A B C D ([Fig. 18]) be the given square. It is required to construct a square which shall be to A B C D as 2 is to 5. Upon the side A B as a diameter describe the semicircle A F B, and divide the line A B into five equal parts. At the second point of division erect the perpendicular E F and join A F; the square described upon A F will be to the given square A B C D as 2 is to 5.
Fig. 19.
To construct, upon a given base, a Rectangle, which shall be similar to a given rectangle.
—Let A E F G ([Fig. 19]) be the given rectangle. It is required to construct upon the base A B, one that shall be similar to A E F G. Produce A E and lay off the given base from A to B; draw the diagonal A G and produce it indefinitely. Erect a perpendicular to A B at B, and from the point D where it intersects the diagonal produced, draw D C perpendicular to A F produced. Then A B C D will be similar to A E F G. All rectangles having their diagonals in the same line are similar.
Fig. 20.
To describe a regular Pentagon on a given line.
—Let A B ([Fig. 20]) be the given line. Bisect A B at C, draw C F perpendicular to A B, and make C D equal to A B. Draw A D and produce it indefinitely; make D E equal to half A B. From A as a centre, with A E as a radius, describe an arc cutting the perpendicular C D in F; and from A F and B as centres, with radius A B, describe arcs cutting each other in G and H; join A G, B H, F G and F H; then A G F H B will be the pentagon required.
Fig. 21.