MathDB

1

Part of 2020 European Mathematical Cup

Problems(2)

Geometry with parallel lines and cyclic quadrilaterals

Source: European Mathematical Cup 2020, Problem J1

12/22/2020
Let ABCABC be an acute-angled triangle. Let DD and EE be the midpoints of sides AB‾\overline{AB} and AC‾\overline{AC} respectively. Let FF be the point such that DD is the midpoint of EF‾\overline{EF}. Let Γ\Gamma be the circumcircle of triangle FDBFDB. Let GG be a point on the segment CD‾\overline{CD} such that the midpoint of BG‾\overline{BG} lies on Γ\Gamma. Let HH be the second intersection of Γ\Gamma and FCFC. Show that the quadrilateral BHGCBHGC is cyclic. \\ \\ Proposed by Art Waeterschoot.
geometrycyclic quadrilateral
concurrency related to parallelogma and a circle

Source: 2020 European Mathematical Cup Seniors P1

12/23/2020
Let ABCDABCD be a parallelogram such that ∣AB∣>∣BC∣|AB| > |BC|. Let OO be a point on the line CDCD such that ∣OB∣=∣OD∣|OB| = |OD|. Let ω\omega be a circle with center OO and radius ∣OC∣|OC|. If TT is the second intersection of ω\omega and CDCD, prove that AT,BOAT, BO and ω\omega are concurrent.
Proposed by Ivan Novak
geometryparallelogramconcurrencyconcurrent