[BLACK-Dragon]

1.In circle quarilateral $ABCD$, a line perpendicular to $AB$ passing through B meets line $CD$ at $B'$. A line perpendicular to $CD$ passing through $D$ meets line $AB$ at $D'$. Prove that $AC$ is parallel to $B'D'$

2.Let $ABC$ be a triangle such that $BC>CA>AB$. Choose points $D$ on $BC$ and $E$ on ray $BA$ such that $BD=BE=AC$. The circumcircle of $\bigtriangleup BED$ intersects $AC$ ay $P $and the line $BP$ intersects the circumcircle of $\bigtriangleup ABC$ again at $Q$ . Prove that $AQ+QC=BP$

3.In $\bigtriangleup ABC$, suppose $D$ is in $AB$ and $E$ in $AC$ such that $BD=DE=EC$ .If $\hat {A} =60^{\circ}$, prove that $BE$ and $CD$ intersects at the circumcircle of $\bigtriangleup ABC$

4. Points $E$ and $F$ are given on side $BC$ of convex quadrilateral $ABCD$(with $E$ closer than $F$ to $B$). If $B \hat {A} E = C\hat {D} F$ and $E \hat {A} F = F \hat {D} E$ , show that $F \hat{A} C= E\hat {D} B$

5. A chord $PQ$ with midpoint $R$ is drawn in a circle with diameter $AB$.Perpendiculars $PS$ and $QT$ are dropped onto $AB$. Prove that $\bigtriangleup RST $ is isosceles