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.