Problem
GEO-B1-M04-P018 Equal Diagonals and Midpoints
#18
★★★☆☆ Level 3 of 5
In quadrilateral \(ABCD\), diagonals meet at \(O\), and \(AO=OC\), \(BO=OD\), \(AC=BD\). Prove that \(ABCD\) is a rectangle.
First prove that it is a parallelogram, then use the equality of diagonals.
From \(AO=OC\) and \(BO=OD\), the diagonals bisect each other. Hence \(ABCD\) is a parallelogram. A parallelogram with equal diagonals has all angles right: this follows by comparing triangles \(ABC\) and \(DCB\). Therefore \(ABCD\) is a rectangle.
A two-step problem: first the type of quadrilateral, then the special case.