2

Part of 2009 Oral Moscow Geometry Olympiad

Problems(2)

intersection point of diagonals of rectangle lies on diagonal of square

Source: 2009 Oral Moscow Geometry Olympiad grades 8-9 p2

A square and a rectangle of the same perimeter have a common corner. Prove that the intersection point of the diagonals of the rectangle lies on the diagonal of the square.
(Yu. Blinkov)

geometrysquarerectangle

gray area is equal to sum of black areas, trapezoid, parallelogram

Source: 2009 Oral Moscow Geometry Olympiad grades 10-11 p2

ABCD

and parallelogram

MBDK

are located so that the sides of the parallelogram are parallel to the diagonals of the trapezoid (see fig.). Prove that the area of the gray part is equal to the sum of the areas of the black part.
(Yu. Blinkov) https://cdn.artofproblemsolving.com/attachments/b/9/cfff83b1b85aea16b603995d4f3d327256b60b.png

geometrytrapezoidareasparallelogram