MathDB

3

Part of 2022 Vietnam TST

Problems(1)

Vietnam TST #3

Source: Vietnam TST 2022 P3

4/26/2022
Let ABCDABCD be a parallelogram, ACAC intersects BDBD at II. Consider point GG inside △ABC\triangle ABC that satisfy ∠IAG=∠IBG≠45∘−∠AIB4\angle IAG=\angle IBG\neq 45^{\circ}-\frac{\angle AIB}{4}. Let E,GE,G be projections of CC on AGAG and DD on BGBG. The E−E- median line of △BEF\triangle BEF and F−F- median line of △AEF\triangle AEF intersects at HH. a)a) Prove that AF,BEAF,BE and IHIH concurrent. Call the concurrent point LL. b)b) Let KK be the intersection of CECE and DFDF. Let JJ circumcenter of (LAB)(LAB) and M,NM,N are respectively be circumcenters of (EIJ)(EIJ) and (FIJ)(FIJ). Prove that EM,FNEM,FN and the line go through circumcenters of (GAB),(KCD)(GAB),(KCD) are concurrent.
geometryparallelogramcircumcircle