RU-ORAL-GEOMETRY 2022

Given an isosceles trapezoid $ABCD$ . The bisector of angle $B$ intersects the base $AD$ at point $L$ . Prove that the center of the circle circumscribed around triangle $BLD$ lies on the circle circumscribed around the trapezoid.

(Yu. Blinkov)

Angle bisectors from vertices $B$ and $C$ and the perpendicular bisector of side $BC$ are drawn in a non-isosceles triangle $ABC$ . Next, three points of pairwise intersection of these three lines were marked (remembering which point is which), and the triangle itself was erased. Restore it according to the marked points using a compass and ruler.

(Yu. Blinkov)

In quadrilateral $ABCD$ , sides $AB$ and $CD$ are equal (but not parallel), points $M$ and $N$ are the midpoints of $AD$ and $BC$ . The perpendicular bisector of $MN$ intersects sides $AB$ and $CD$ at points $P$ and $Q$ , respectively. Prove that $AP = CQ$ .

(M. Kungozhin)