Prove that if the diagonals of a parallelogram are perpendicular then such a parallelogram is a rhombus.

univerkov | April 29, 2021 | education

Since ABCD is a parallelogram, its diagonals, at the point of their intersection, are divided in half.

Then OB = OD, OA = OC.

By condition, BD is perpendicular to AC, then triangles AOB, BOC, COD, AOD are rectangular and equal in two legs.

Then AB = BC = CD = AD.

A parallelogram in which all sides are equal is a rhombus, as required.

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