26. If a semicircle and its diameter be touched by any circle, either internally or externally, twice the rectangle contained by the radius of the semicircle, and the radius of the tangential circle, is equal to the rectangle contained by the segments of any secant to the semicircle, through the point of contact of the diameter and touching circle. 27. If ρ, ρ′ be the radii of two circles, touching each other at the centre of the inscribed circle of a triangle, and each touching the circumscribed circle, prove and state and prove corresponding theorems for the escribed circles. 28. If from any point in the circumference of the circle, circumscribed about a regular polygon of n sides, lines be drawn to its angular points, the sum of their squares is equal to 2n times the square of the radius. 29. In the same case, if the lines be drawn from any point in the circumference of the inscribed circle, prove that the sum of their squares is equal to n times the sum of the squares of the radii of the inscribed and the circumscribed circles. 30. State the corresponding theorem for the sum of the squares of the lines drawn from any point in the circumference of any concentric circle. 31. If from any point in the circumference of any concentric circle perpendiculars be let fall on all the sides of any regular polygon, the sum of their squares is constant. 32. For the inscribed circle, the constant is equal to times the square of the radius. 33. For the circumscribed circle, the constant is equal to n times the square of the radius of the inscribed circle, together with n times the square of the radius of the circumscribed circle. 34. If the circumference of a circle whose radius is R be divided into seventeen equal parts, and AO be the diameter drawn from one of the points of division (A), and if ρ1, ρ2……ρ8 denote the chords from O to the points of division, A1, A2……A8 on one side of AO, then Dem.—Let the supplemental chords corresponding to ρ1, ρ2, &c., be denoted by r1, r2, &c.; then [III. xxxv. Ex. 2], we have ρ1r1 = Rr2, ρ2r2 = Rr4, ρ4r4 = Rr8, ρ8r8 = Rr1, Hence ρ1ρ2ρ4ρ8 = R4. And it may be proved in the same manner that ρ1ρ2ρ3ρ4ρ5ρ6ρ7ρ8 = R8. Therefore ρ3ρ5ρ6ρ7 = R4. 35. If from the middle point of the line joining any two of four concyclic points a perpendicular be let fall on the line joining the remaining two, the six perpendiculars thus obtained are concurrent. 36. The greater the number of sides of a regular polygon circumscribed about a given circle, the less will be its perimeter. 37. The area of any regular polygon of more than four sides circumscribed about a circle is less than the square of the diameter. 38. Four concyclic points taken three by three determine four triangles, the centres of whose nine-points circles are concyclic.