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circle of diameter OI is tangent to 2 circles, Gauss line of cyclic ABCD

Source: 2021 Saudi Arabia Training Lists p21 https://artofproblemsolving.com/community/c2758131_2021_saudi_arabia_training_tests

January 6, 2022
cyclic quadrilateralgeometrytangent

Problem Statement

Let ABCDABCD be a cyclic quadrilateral with OO is circumcenter and ACAC meets BDBD at II Suppose that rays DA,CDDA,CD meet at EE and rays BA,CDBA,CD meet at FF. The Gauss line of ABCDABCD meets AB,BC,CD,DAAB,BC,CD,DA at points M,N,P,QM,N,P,Q respectively. Prove that the circle of diameter OIOI is tangent to two circles (ENQ),(FMP)(ENQ), (FMP)
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