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.