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(ASG) tangent to (J)

Source: 6th XMO (China) p5 https://artofproblemsolving.com/community/c3192772_geometry_regional_china

April 7, 2024
tangent circlesgeometry

Problem Statement

As shown in the figure, ⊙O\odot O is the circumcircle of △ABC\vartriangle ABC, ⊙J\odot J is inscribed in ⊙O\odot O and is tangent to ABAB, ACAC at points DD and E respectively, line segment FGFG and ⊙O\odot O are tangent to point AA, and AF=AG=ADAF =AG=AD, the circumscribed circle of △AFB\vartriangle AFB intersects ⊙J\odot J at point SS. Prove that the circumscribed circle of △ASG\vartriangle ASG is tangent to ⊙J\odot J. https://cdn.artofproblemsolving.com/attachments/a/a/62d44e071ea9903ebdd68b43943ba1d93b4138.png
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