MathDB

2

Part of 2005 Poland - Second Round

Problems(2)

Poland second round 2005

Source:

2/18/2006
In a convex quadrilateral ABCDABCD, point MM is the midpoint of the diagonal ACAC. Prove that if ∠BAD=∠BMC=∠CMD\angle BAD=\angle BMC=\angle CMD, then a circle can be inscribed in quadrilateral ABCDABCD.
geometry unsolvedgeometry
In the rhombus ABCD

Source:

12/6/2010
A rhombus ABCDABCD with ∠BAD=60∘\angle BAD=60^{\circ} is given. Points EE on side ABAB and FF on side ADAD are such that ∠ECF=∠ABD\angle ECF=\angle ABD. Lines CECE and CFCF respectively meet line BDBD at PP and QQ. Prove that PQEF=ABBD\frac{PQ}{EF}=\frac{AB}{BD}.
geometryrhombustrapezoidgeometry proposed