5. If a square piece of wood be divided into four equal squares, of which one is removed, prove that the remaining gnomon may be made into a square by cutting it into four parts only.* 6. If CD be drawn bisecting the angle C of the triangle ABC; and AB be produced to E, a point equidistant from C and D; prove that the rectangle AE.EB is equal to the square of ED. 7. The square inscribed in a semicircle is to the square inscribed in the circle as 2: 5.

8. The three straight lines drawn from the angular points of a triangle perpendicular to the opposite sides, bisect the angles of the triangle which is formed by joining the points in which they meet the sides of the original triangle.

9. If on a given finite straight line a semicircle and a quadrant be described, the area of the lune which is contained between them is equal to that of the triangle whose base is the given straight line, and its vertex the centre of the circle of which the quadrant is a part.

10. To divide a given circle into any number of equal parts by means of concentric circles.

11. If the diameter of a circle be divided into any number of equal parts, and a series of semicircles be described on one side of the diameter, all passing through one extremity of it, and having for their diameters respectively, one, two, &c., of the equal parts, and the same be done at the other extremity of the diameter on the other side of it, the circle shall itself be divided into the same number of equal parts, and the containing arcs shall be of equal length. 12. If on the three sides of a right-angled triangle three semicircles be described, the area of the triangle is equal to the sum of the areas of the two lunes inclosed by the circles.

13. If from the semi-perimeter of a triangle there be subtracted successively two sides, the rectangle by the remainders is equal to the rectangle by the radii of the two circles which touch respectively the base and the two sides, and the base and the two sides produced.

14. If on the three sides of a triangle, three equilateral triangles be described, the straight lines which join their three centres of gravity (Book I., Ded. 55) shall form an equilateral triangle. 15. If from any point in a circular arc perpendiculars be drawn to its bounding radii, the distance of the points at which they meet the radii is always the same.

*This Problem may be solved by five-and-twenty different methods.

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THE END.