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In-the-square-PQRS-K-is-the-midpoint-of-PQ-L-is-the-midpoint-of-QR-M-is-the-midpoint-RS-N-is-the-midpoint-of-SP-and-O-is-the-midpoint-of-KM-A-line-segment-is-drawn-from-each-pair-of-points-from-

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Question Number 110523 by Aina Samuel Temidayo last updated on 29/Aug/20
In the square PQRS, K is the midpoint of PQ, L is the midpoint of QR, M is the midpoint RS, N is the midpoint of SP and O is the midpoint of KM. A line segment is drawn from each pair of points from (K,L,M,N,O,P,Q,R,S). These line segments create points of intersections not contained in (K,L,M,N,O,P,Q,R,S). How many distinct such points are there?
$$ \\ $$$$\mathrm{In}\:\mathrm{the}\:\mathrm{square}\:\mathrm{PQRS},\:\mathrm{K}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{midpoint}\:\mathrm{of}\:\mathrm{PQ},\:\mathrm{L}\:\mathrm{is}\:\mathrm{the}\:\mathrm{midpoint}\:\mathrm{of} \\ $$$$\mathrm{QR},\:\mathrm{M}\:\mathrm{is}\:\mathrm{the}\:\mathrm{midpoint}\:\mathrm{RS},\:\mathrm{N}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{midpoint}\:\mathrm{of}\:\mathrm{SP}\:\mathrm{and}\:\mathrm{O}\:\mathrm{is}\:\mathrm{the}\:\mathrm{midpoint} \\ $$$$\mathrm{of}\:\mathrm{KM}.\:\mathrm{A}\:\mathrm{line}\:\mathrm{segment}\:\mathrm{is}\:\mathrm{drawn}\:\mathrm{from} \\ $$$$\mathrm{each}\:\mathrm{pair}\:\mathrm{of}\:\mathrm{points}\:\mathrm{from} \\ $$$$\left(\mathrm{K},\mathrm{L},\mathrm{M},\mathrm{N},\mathrm{O},\mathrm{P},\mathrm{Q},\mathrm{R},\mathrm{S}\right).\:\mathrm{These}\:\mathrm{line} \\ $$$$\mathrm{segments}\:\mathrm{create}\:\mathrm{points}\:\mathrm{of} \\ $$$$\mathrm{intersections}\:\mathrm{not}\:\mathrm{contained}\:\mathrm{in} \\ $$$$\left(\mathrm{K},\mathrm{L},\mathrm{M},\mathrm{N},\mathrm{O},\mathrm{P},\mathrm{Q},\mathrm{R},\mathrm{S}\right).\:\mathrm{How}\:\mathrm{many} \\ $$$$\mathrm{distinct}\:\mathrm{such}\:\mathrm{points}\:\mathrm{are}\:\mathrm{there}? \\ $$

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