Midsegment Theorem and Coordinate Proof

Midsegment of a triangle is a segment that connects the midpoints of two sides of the triangle. Every triangle has three (3) midsegments.

Note that point D, E, and F are midpoints of their respective segments.

If we connect each midpoint by line segments, we obtain all three midpoint segments.

The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that side.
Notice, by the midsegment theorem,

Example: Length of MidsegmentsNote that A and B are midpoints. By the Midsegments Theorem we know that segment AB  is 1/2 the length of segment XZ. 

Roof Trusses

 In the kaleidoscope image, $\overline{AE}\cong\overline{BE}$ and  $\overline{AD}\cong\overline{CD}.$  Show that  $\overline{CB}\parallel\overline{DE}.$  

Because $\overline{AE}\cong\overline{BE}$ and  $\overline{AD}\cong\overline{CD,}\ E$ is the midpoint of  $\overline{AB}$ and $D$ is the midpoint of  $\overline{AC}$  by definition.  Then  $\overline{DE}$  is a midsegment of   $\Delta ABC$  by definition and  $\overline{CB}\parallel\overline{DE}$  by the Midsegment Theorem.