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Vietnam TST #4

Source: Vietnam TST 2022 P4

April 27, 2022
geometrycircumcirclegeometric transformationreflection

Problem Statement

An acute, non-isosceles triangle ABCABC is inscribed in a circle with centre OO. A line go through OO and midpoint II of BCBC intersects AB,ACAB, AC at E,FE, F respectively. Let D,GD, G be reflections to AA over OO and circumcentre of (AEF)(AEF), respectively. Let KK be the reflection of OO over circumcentre of (OBC)(OBC). a)a) Prove that D,G,KD, G, K are collinear. b)b) Let M,NM, N are points on KB,KCKB, KC that IM⊥ACIM\perp AC, IN⊥ABIN\perp AB. The midperpendiculars of IKIK intersects MNMN at HH. Assume that IHIH intersects AB,ACAB, AC at P,QP, Q respectively. Prove that the circumcircle of △APQ\triangle APQ intersects (O)(O) the second time at a point on AIAI.
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