In trapezoid ABCD, point K is the Midpoint of the Base AB. It is known that CK = KD. Prove that the trapezoid is isosceles.

Consider the ACD triangle, in which, by condition, KD = KС, then the ACD triangle is isosceles, then its angles at the base are equal. KDС angle = KСD.

In the ADK triangle, the angle AKD = КDС as the criss-crossing angles at the intersection of parallel lines AB and СD of the secant КD. Similarly, the angle CКВ = KСD.

Then the angle AKD = ВKС = KDС = KСD.

In triangles AKD and ВKС AK = ВK, DK = СK, the angle AKD = ВKС, therefore, the triangles are equal on two sides and the angle between them.

Then AD= BC, and consequently, the trapezoid of AВСD is isosceles.

Q.E.D.