Dividing a circle into areas lines (there are two points on one side and one on the other). So the recurrence can be expressed as which can be easily reduced to using the sums of the first formula_3 natural numbers and the first formula_3 squares, this combines to Finally which yields The lemma asserts that the number of regions is maximal if all "inner" intersections of chords are simple (exactly two chords pass through each point of intersection in the interior). This will be the case if the points on the circle are chosen "in general position". Under this assumption of "generic intersection", the number
Fife Circle Line the full circle to Glenrothes. Some services regularly ran through to/from until 2015, but with the opening of the Borders Railway in September that year this routing ceased (except for a small number of weekday peak trains). Here the main line and loop line divide. The two lines join forming a circle. The east peninsula of Fife beyond Kirkcaldy is not served by railways post-Beeching. At one time a railway ran all the way from Thornton to St Andrews round the coast of Fife. The Beeching Report in fact recommended services continue between Thornton and Leven but these fell victim