Problem
GEO-B1-M04-P007 Diagonals Bisect Each Other
#7
★★☆☆☆ Level 2 of 5
In quadrilateral \(ABCD\), the diagonals meet at \(O\), with \(AO=OC\) and \(BO=OD\). Prove that \(ABCD\) is a parallelogram without citing the criterion directly.
Compare triangles \(AOB\) and \(COD\), then \(AOD\) and \(COB\).
Triangles \(AOB\) and \(COD\) are congruent by SAS: \(AO=OC\), \(BO=OD\), and angles \(AOB\) and \(COD\) are vertical. Hence \(\angle ABO=\angle CDO\), so \(AB\parallel CD\). Similarly, triangles \(AOD\) and \(COB\) are congruent, so \(AD\parallel BC\). Both pairs of opposite sides are parallel, hence \(ABCD\) is a parallelogram.
Useful as a proof of the criterion itself, not just as its application.