ABCD is a trapezoid, point O is the point of intersection of its diagonals

ABCD is a trapezoid, point O is the point of intersection of its diagonals, equidistant from the lateral sides AB and CD. Prove that the trapezoid is isosceles.

By the property of the trapezoid diagonals, they divide the trapezoid into four triangles, two of which belong to the lateral sides, are equal.

Saov = Ssod.

By condition, OK is perpendicular to AB, OH is perpendicular to CH, therefore these are the heights of the triangles.

Also, by condition, OK = OH, then:

Saov = AB * OK / 2.

Sod = CD * OH / 2.

AB * OK / 2 = CD * OH / 2.

And since OK = OH, then AB = CD, which was required to be proved.