What condition is necessary on the point D for this to work?

Proof: Let ABC be any triangle. Let D be a point on [math]\overline{AC}[/math]. Decompose this triangle into four sections are shown above. Using isometries, we can rearrange these shapes into a trapezoid. In order for the decomposed shapes to form a trapezoid, point D must be equivalent to point I or [math]\overline{BH}\cong\overline{CD}[/math]. When this occurs, each of the sides are equal in length. This allows for a set of parallel lines to exist which satisfies the definition of a trapezoid. [math]\diamondsuit[/math]