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.
39. If two sides of a triangle be given in position, and if their included angle be equal to an angle of an equilateral triangle, the locus of the centre of its nine-points circle is a right line.
40. If, in the hypothesis and notation of Ex. 34, α, β denote any two suffixes whose sum is less than 8, and of which α is the greater,
For instance, ρ1ρ4 = R(ρ3 + ρ5) [III. xxxv., Ex. 7].
In the same case, if the suffixes be greater than 8,
For instance, ρ8ρ2 = R(ρ6 − ρ7) [III. xxxv., Ex. 6].