3. The area of a regular hexagon inscribed in a circle is half the area of an equilateral triangle, and three-fourths of the area of a regular hexagon circumscribed to the circle. PROP. XVI.—Problem. To inscribe a regular polygon of fifteen sides in a given circle. Sol.—Inscribe a regular pentagon ABCDE in the circle [xi.], and also an equilateral triangle AGH [ii.]. Join CG. CG is a side of the required polygon. Dem.—Since ABCDE is a regular pentagon, the arc ABC is ths of the circumference; and since AGH is an equilateral triangle, the arc ABG is rd of the circumference. Hence the arc GC, which is the difference between these two arcs, is equal to ths −rd, or th of the entire circumference; and therefore, if chords equal to GC [i.] be placed round the circle, we shall have a regular polygon of fifteen sides, or quindecagon, inscribed in it. Scholium.—Until the year 1801 no regular polygon could be described by constructions employing the line and circle only, except those discussed in this Book, and those obtained from them by the continued bisection of the arcs of which their sides are the chords; but in that year the celebrated Gauss proved that if 2n + 1 be a prime number, regular polygons of 2n + 1 sides are inscriptable by elementary geometry. For the case n = 4, which is the only figure of this class except the pentagon for which a construction has been given, see Note at the end of this work. Questions for Examination on Book IV. 1. What is the subject-matter of Book IV.? 2. When is one rectilineal figure said to be inscribed in another? 3. When circumscribed? 4. When is a circle said to be inscribed in a rectilineal figure? 5. When circumscribed about it? 6. What is meant by reciprocal propositions? Ans. In reciprocal propositions, to every line in one there corresponds a point in the other; and, conversely, to every point in one there corresponds a line in the other. 7. Give instances of reciprocal propositions in Book IV. 8. What is a regular polygon? 9. What figures can be inscribed in, and circumscribed about, a circle by means of Book IV.? 10. What regular polygons has Gauss proved to be inscriptable by the line and circle? 11. What is meant by escribed circles? 12. How many circles can be described to touch three lines forming a triangle? 13. What is the centroid of a triangle? 14. What is the orthocentre? 15. What is the circumcentre? 16. What is the polar circle? 17. When is the polar circle imaginary? 18. What is the “nine-points circle”? 19. Why is it so called? 20. Name the special nine points through which it passes. 21. What three regular figures can be used in filling up the space round a point? Ans. Equilateral triangles, squares, and hexagons. 22. If the sides of a triangle be 13, 14, 15, what are the values of the radii of its inscribed and escribed circles?