# In isosceles trapezoid ABCD, point E is the midpoint of the larger base AD

In isosceles trapezoid ABCD, point E is the midpoint of the larger base AD, ED = EC, angle BAD = 60 degrees, prove that quadrilateral ABCE is a rhombus.

Since the trapezoid is isosceles, the angle CDA, at the base, is equal to the angle BAD and is equal to 60.

Consider a triangle CED, in which, according to the condition, ED = EC, that is, an isosceles triangle, then the angle ECD = CDA = 60. Since the two angles are equal to 60, then the angle CED = 60.

Then the triangle CED is equilateral.

Since the angles BAD and CED are equal to each other, AB is parallel to EC, and since the bases of the trapezoid AD and BC are also parallel, then AE is parallel to BC, hence ABCE is a parallelogram.

Since point E is the middle of AD, then AE = AD = EC = AB, therefore, ABCE is a rhombus.

Q.E.D.

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