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hexagon ABCDEF are each of them equal to the angle AFE or FED; therefore
3. The hexagon ABCDEF is equiangular;
and it is equilateral, as was shown; and it is inscribed in the given circle ABCDEF. Q.E.F.
COR. From this it is manifest, that the side of the hexagon is equal to the straight line from the centre, that is, to the semidiameter of the circle.
And if through the points A, B, C, D, E, F, there be drawn straight lines touching the circle, an equilateral and equiangular hexagon shall be described about the circle, which may be demonstrated from what has been said of the pentagon; and likewise a circle may be inscribed in a given equilateral and equiangular hexagon, and circumscribed about it, by a method like to that used for the pentagon.
To inscribe an equilateral and equiangular quindecagon in a given circle. Let ABCD be the given circle; it is required to inscribe an equilateral and equiangular quindecagon in the circle ABCD.
Let AC be the side of an equilateral triangle inscribed (IV. 2.) in the circle, and AB the side of an equilateral and equiangular pentagon inscribed (IV. 11.) in the same; therefore
1. Of such equal parts as the whole circumference ABCDF contains fifteen, the circumference ABC contains five, it being the third part of the whole, and
2. The circumference AB contains three parts, it being the fifth part of the whole; therefore
3. Their difference BC contains two of the same parts. Bisect (III. 30.) BC in E; therefore
4. BE, EC, are, each of them, the fifteenth part of the whole circumference ABCD;
therefore if the straight lines BE, EC, be drawn, and straight lines equa to them be placed around (IV. 1.) in the whole circle, an equilateral and equiangular quindecagon shall be inscribed in it.
And in the same manner as was done in the pentagon, if, through the points of division made by inscribing the quindecagon, straight lines be drawn touching the circle, an equilateral and equiangular quindecagon shall be described about the circle; and likewise, as in the pentagon, a circle may be inscribed in a given equilateral and equiangular quindecagon, and circumscribed about it.
1. On a given straight line to describe an isosceles triangle that shall have each of its equal sides double of the base.
2. The line which bisects the vertical angle of an isosceles triangle, bisects the base perpendicularly.
3. To describe an isosceles triangle, the base and one of the sides being given.
4. The difference between two sides of a triangle is less than the third side.
5. To trisect a right angle, that is, to divide it into three equal parts. 6. To trisect a given finite straight line.
7. If the vertical angle of an isosceles triangle be a right angle, each of the angles at the base is half a right angle.
8. In the diagram Book I. Prop. 5. draw AH to the point H in which CF and BG intersect: show that CH is equal to BH; FH to GH; and that the angle BAC is bisected by AH.
9. The angles of a quadrilateral are equal to four right angles.
10. To describe a square equal to the difference of two given squares.
11. A line joining the middle points of two sides of a triangle is parallel to the base, and equal to half of it.
12. To bisect a given parallelogram by a line drawn from a point in one of its sides.
13. To bisect a given triangle by a line drawn from a point in one of its sides.
14. To divide a given straight line into two parts, so that their rectangle may be equal to a given square.
15. To produce a given straight line, so that the rectangle of the whole line and the given line may be equal to a given square.
16. The square on the base of an isosceles triangle, whose vertical angle is a right angle, is equal to four times the area of the triangle.
17. The squares of the diagonals of a parallelogram are equal to the sum of the squares of the four sides.
18. With a given radius to describe a circle which shall pass through two given points.
19. The arcs intercepted between any two parallel chords in a circle are equal.
20. From one extremity of a line which cannot be produced, to draw a line perpendicular to it.
21. Given the vertical angle, the base, and the altitude of a triangle, to construct it.
22. To divide a circle into two parts, such that the angle contained in one segment shall equal twice the angle contained in the other.
23. If the diameter of a circle be one of the equal sides of an isosceles triangle, the base will be bisected by the circumference.
24. To describe a circle which shall pass through a given point, and which shall touch a given straight line in a given point.
25. To describe a circle which shall touch a given straight line, and pass through two given points.
26. To trisect a given circle by dividing it into three equal sectors.
27. To describe an equilateral triangle about a square.
28. To draw a tangent to a given circle parallel to a given straight line.
29. To draw a tangent to a given circle making a given angle with a given straight line.
30. Find algebraical formulæ corresponding to the enunciations of the Second Book.
31. A flag-staff of a given height is erected on a tower, the height of which is also given; at what point on the horizontal line drawn from the foot of the tower will the flag-staff appear under the greatest angle?