Let there be 9 fixed points on the circumference of a circle . Each of these points is joined to every one of the remaining 8 points by a straight line and the points are so positioned on the circumfe

Question

Let there be 9 fixed points on the circumference of a circle . Each of these points is joined to every one of the remaining 8 points by a straight line and the points are so positioned on the circumference that atmost 2 straight lines meet in any interior point of the circle. The number of such interior intersection points is

TestHub · Accepted Answer

Any interior intersection point corresponds to 4 of the fixed points , namely the 4 end points of the intersecting segments . Conversely, any 4 labled points determine a quadrilateral, the diagonals of which intersect once within the circle.

Number of interior intersection points $=^{9} C_{4} = 126$

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