MathDB

3

Part of 2022 Oral Moscow Geometry Olympiad

Problems(2)

AP = CQ wanted, axis of MN related, midpoints of AD,BC where AB=CD

Source: 2022 Oral Moscow Geometry Olympiad grades 8-9 p3

4/17/2022
In quadrilateral ABCDABCD, sides ABAB and CDCD are equal (but not parallel), points MM and NN are the midpoints of ADAD and BCBC. The perpendicular bisector of MNMN intersects sides ABAB and CDCD at points PP and QQ, respectively. Prove that AP=CQAP = CQ.
(M. Kungozhin)
equal segmentsgeometry
intersection of diagonals of parallelogram lies on a diagonal of convex ABCD

Source: 2022 Oral Moscow Geometry Olympiad grades 10-11 p3

4/18/2022
Extensions of opposite sides of a convex quadrilateral ABCDABCD intersect at points PP and QQ. Points are marked on the sides of ABCDABCD (one per side), which are the vertices of a parallelogram with a side parallel to PQPQ. Prove that the intersection point of the diagonals of this parallelogram lies on one of the diagonals of quadrilateral ABCDABCD.
(E. Bakaev)
geometryparallelogramcollinear