23. What is the radius of the circumscribed circle? 24. What is the radius of its nine-points circle? 25. What is the distance between the centres of its inscribed and circumscribed circles? 26. If r be the radius of a circle, what is the area of its inscribed equilateral triangle?—of its inscribed square?—its inscribed pentagon?—its inscribed hexagon?—its inscribed octagon?—its inscribed decagon? 27. With the same hypothesis, find the sides of the same regular figures. Exercises on Book IV. 1. If a circumscribed polygon be regular, the corresponding inscribed polygon is also regular, and conversely. 2. If a circumscribed triangle be isosceles, the corresponding inscribed triangle is isosceles, and conversely. 3. If the two isosceles triangles in Ex. 2 have equal vertical angles, they are both equilateral. 4. Divide an angle of an equilateral triangle into five equal parts. 5. Inscribe a circle in a sector of a given circle. 6. The line DE is parallel to the base BC of the triangle ABC: prove that the circles described about the triangles ABC, ADE touch at A. 7. The diagonals of a cyclic quadrilateral intersect in E: prove that the tangent at E to the circle about the triangle ABE is parallel to CD. 8. Inscribe a regular octagon in a given square. 9. A line of given length slides between two given lines: find the locus of the intersection of perpendiculars from its extremities to the given lines. 10. If the perpendicular to any side of a triangle at its middle point meet the internal and external bisectors of the opposite angle in the points D and E; prove that D, E are points on the circumscribed circle. 11. Through a given point P draw a chord of a circle so that the intercept EF may subtend a given angle X. 12. In a given circle inscribe a triangle having two sides passing through two given points, and the third parallel to a given line. 13. Given four points, no three of which are collinear; describe a circle which shall be equidistant from them. 14. In a given circle inscribe a triangle whose three sides shall pass through three given points. 15. Construct a triangle, being given— The radius of the inscribed circle, the vertical angle, and the perpendicular from the vertical angle on the base. The base, the sum or difference of the other sides, and the radius of the inscribed circle, or of one of the escribed circles. The centres of the escribed circles. 16. If F be the middle point of the base of a triangle, DE the diameter of the circumscribed circle which passes through F, and L the point where a parallel to the base through the vertex meets DE: prove DL.FE is equal to the square of half the sum, and DF.LE equal to the square of half the difference of the two remaining sides.