MathDB
Contrived geo strikes again

Source: Vietnam TST 2023 P5

April 14, 2023
geometry

Problem Statement

Let ABCDABCD be a convex quadrilateral with ∠B<∠A<90o\angle B < \angle A < 90^{o}. Let II be the midpoint of ABAB and SS the intersection of ADAD and BCBC. Let RR be a variable point inside the triangle SABSAB such that ∠ASR=∠BSR\angle ASR = \angle BSR. On the straight lines AR,BRAR, BR , take the points E,FE, F, respectively so that BE,AFBE , AF are parallel to RSRS. Suppose that EFEF intersects the circumcircle of triangle SABSAB at points H,KH, K. On the segment ABAB, take points M,NM , N such that ∠AHM=∠BHI\angle AHM =\angle BHI , ∠BKN=∠AKI\angle BKN = \angle AKI.
a) Prove that the center JJ of the circumcircle of triangle SMNSMN lies on a fixed line.
b) On BE,AFBE, AF , take the points P,QP, Q respectively so that CPCP is parallel to SESE and DQDQ is parallel to SFSF. The lines SE,SFSE, SF intersect the circle (SAB)(SAB), respectively, at U,VU, V. Let GG be the intersection of AUAU and BVBV. Prove that the median of vertex GG of the triangle GPQGPQ always passes through a fixed point .
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