BMO shortlist 2011

[ghenghea]

Let $ABCD$ be a cyclic quadrilateral such that the lines $BC$ and $AD$ intersect in $P$.Let $Q $ be a point on line $BP$,different from $B$,such that $BP=PQ$.
Consider the parallelograms $CAQR$ and $DBCS$.
Prove that the points $C,Q,R,S$ all lie on the same circle.

[Sergic Primazon]

ghenghea в сообщении #1181413 писал(а):

Let $ABCD$ be a cyclic quadrilateral such that the lines $BC$ and $AD$ intersect in $P$.Let $Q $ be a point on line $BP$,different from $B$,such that $BP=PQ$.
Consider the parallelograms $CAQR$ and $DBCS$.
Prove that the points $C,Q,R,S$ all lie on the same circle.

$AP=PA_1\ ,\ CP=PC_1 \Rightarrow BDC_1A_1- cyclic$

$\overrightarrow{BC}=\overrightarrow{DS}=\overrightarrow{C_1Q}=\overrightarrow{A_1R}$