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(HRS) tangent to OD wanted, circumcircle related

Source: 2021 Saudi Arabia Training Lists p19 https://artofproblemsolving.com/community/c2758131_2021_saudi_arabia_training_tests

January 6, 2022
geometrytangentcircumcircle

Problem Statement

Let ABCABC be a triangle with AB<ACAB < AC inscribed in (O)(O). Tangent line at AA of (O)(O) cuts BCBC at DD. Take HH as the projection of AA on ODOD and E,FE,F as projections of HH on AB,ACAB,AC.Suppose that EFEF cuts (O)(O) at R,SR,S. Prove that (HRS)(HRS) is tangent to ODOD
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