The fact that the trapezoid is isosceles is actually irrelevant. Since and are the midpoints of legs and , it holds by definition that

and

It follows that

However, the bases of each trapezoid are noncongruent, by definition, so, in particular,

Assume that is the longer base - this argument works symmetrically if the opposite is true. Let - equivalently, .

Since the bases are of unequal length, . The length of the midsegment is the arithmetic mean of the lengths of these two bases, so

Since , it follows that

,

and

.

This disproves similarity, since one condition is that corresponding sides must be in proportion.