MathDB

6

Part of 2021 South East Mathematical Olympiad

Problems(2)

China South East Mathematical Olympiad 2021 Grade10 P6

Source:

7/30/2021
Let ABCDABCD be a cyclic quadrilateral. Let EE be a point on side BC,BC, FF be a point on side AE,AE, GG be a point on the exterior angle bisector of ∠BCD,\angle BCD, such that EG=FG,EG=FG, ∠EAG=12∠BAD.\angle EAG=\dfrac12\angle BAD. Prove that AB⋅AF=AD⋅AE.AB\cdot AF=AD\cdot AE.
geometrycyclic quadrilateral
China South East Mathematical Olympiad 2021 Grade11 P6

Source:

7/30/2021
Let ABCDABCD be a cyclic quadrilateral. The internal angle bisector of ∠BAD\angle BAD and line BCBC intersect at E.E. MM is the midpoint of segment AE.AE. The exterior angle bisector of ∠BCD\angle BCD and line ADAD intersect at F.F. The lines MFMF and ABAB intersect at G.G. Prove that if AB=2AD,AB=2AD, then MF=2MG.MF=2MG.
geometrycyclic quadrilateral