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Magical conditional geometry

Source: India EGMO 2022 TST P3

November 21, 2021

Problem Statement

Let II be incentre of scalene △ABC\triangle ABC and let LL be midpoint of arc BACBAC. Let MM be midpoint of BCBC and let the line through MM parallel to AIAI intersect LILI at point PP. Let QQ lie on BCBC such that PQ⊥LIPQ\perp LI. Let SS be midpoint of AMAM and TT be midpoint of LILI. Prove that IS⊥BCIS\perp BC if and only if AQ⊥STAQ\perp ST.
~Mahavir Gandhi
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