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Geometry Mathley 9.3 concyclic, collinear midpoints

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June 7, 2020
collinearConcyclicgeometrycyclic quadrilateral

Problem Statement

Let ABCDABCD be a quadrilateral inscribed in a circle (O)(O). Let (O1),(O2),(O3),(O4)(O_1), (O_2), (O_3), (O_4) be the circles going through (A,B),(B,C),(C,D),(D,A)(A,B), (B,C),(C,D),(D,A). Let X,Y,Z,TX, Y,Z, T be the second intersection of the pairs of the circles: (O1)(O_1) and (O2),(O2)(O_2), (O_2) and (O3),(O3)(O_3), (O_3) and (O4),(O4)(O_4), (O_4) and (O1)(O_1). (a) Prove that X,Y,Z,TX, Y,Z, T are on the same circle of radius II. (b) Prove that the midpoints of the line segments O1O3,O2O4,OIO_1O_3,O_2O_4,OI are collinear.
Nguyễn Văn Linh
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