### Video Transcript

In the given figure, πΈ and πΉ are
the midpoints of line segments π΄π΅ and π΄πΆ, respectively, π΅π· equals one-half
π΅πΆ, and π΅ lies on line segment π·πΆ. What is the shape of πΈπΉπ΅π·?

We can begin this question by
noting that the diagram shows the information that we have line segments that are
divided into two congruent pieces, since πΈ is the midpoint of line segment π΄π΅ and
πΉ is the midpoint of line segment π΄πΆ. The fact that we have these two
midpoints might indicate that one of the triangle midsegment theorems can be
applied: this one in particular that the line segment joining the midpoints of two
sides of a triangle is parallel to the third side and is half its length.

So, if we consider triangle π΄π΅πΆ,
line segment πΈπΉ is a line segment connecting the midpoints of two sides of a
triangle. Therefore, it is parallel to the
third side, which is π΅πΆ, and πΈπΉ must be half the length of this side π΅πΆ. And notice that we are given
another side which is also equal to one-half π΅πΆ: this line segment, π΅π·.

So now, if we consider the
quadrilateral πΈπΉπ΅π·, we know that this quadrilateral has a pair of congruent
sides. And we can also say that these two
sides are parallel, because we determined that line segment πΈπΉ was parallel to
line segment π΅πΆ. And we know that π΅ lies on the
line segment π·πΆ.

Now, the shape πΈπΉπ΅π· does look
like a parallelogram, and in fact what we have shown here would prove this, because
one way we can prove a quadrilateral is a parallelogram is by showing that one pair
of opposite sides in a quadrilateral are both parallel and congruent. Therefore, we can give the answer
that πΈπΉπ΅π· is a parallelogram.