50. Inscribe in a given circle a triangle having its three sides parallel to three given lines.
51. If the sides AB, BC, &c., of a regular pentagon be bisected in the points A′, B′, C′, D′, E′, and if the two pairs of alternate sides, BC, AE; AB, DE, meet in the points A′′, E′′, respectively, prove
52. In a circle, prove that an equilateral inscribed polygon is regular, and also an equilateral circumscribed polygon, if the number of sides be odd.
53. Prove also that an equiangular circumscribed polygon is regular, and an equiangular inscribed polygon, if the number of sides be odd.
54. The sum of the perpendiculars drawn to the sides of an equiangular polygon from any point inside the figure is constant.
55. Express the sides of a triangle in terms of the radii of its escribed circles.
BOOK V.
THEORY OF PROPORTION
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DEFINITIONS.