SOLUTION: In a triangle, a segment is drawn joining the midpoint of two sides. What is true about this segment?

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Question 748157: In a triangle, a segment is drawn joining the midpoint of two sides. What is true about this segment?
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!
The straight line connecting midpoints of two sides of a triangle is to the side of the triangle.
The straight line connecting midpoints of two sides of a triangle is parallel to the third side of the triangle.

Proof
Figure 1 shows the triangle ABC with the midpoints D and E that are                 
located in its sides BC and AC respectively. The theorem states that
the straight line ED, which connects the midpoints D and E (green
line in the Figure 1), is parallel to the triangle side AB.

Continue the straight line segment ED to its own length to the point F
(Figure 2) and connect the points B and F by the straight line segment BF.
The triangles EDC and FDB have the congruent vertical angles EDC and
FDB, congruent sides DC and DB as halves of the side BC, and congruent
sides ED and FD by the construction. Therefore, these triangles are
congruent in accordance to the postulate P1 (SAS) of the lesson
Congruence tests for triangles (which is under the topic Triangles
in the section Geometry in this site).

Figure 1. To the Theorem 1      

Figure 2. To the proof of the Theorem 1

This means that the angles ECD and DBF are congruent as the corresponding angles of these triangles.
Hence, the straight lines AC and BF are parallel, because these angles are the alternate interior angles formed by the transverse line BC
(see the lesson Parallel lines under the topic Angles, complementary, supplementary angles in the section Geometry in this site).
This means also that the segments CE and BF are of equal length as the corresponding sides of triangles EDC and FDB.
Since the point E is the midpoint of the side AB and the segments AE and CE are of equal length, this implies that the segments BF and AE are of equal length.
Thus, we have proved that in the quadrilateral ADFE the two opposite sides BF and AE are parallel and have equal length.

At this point we can refer to the geometry fact proven in the lesson Properties of the sides of parallelograms (see the Theorem 1 of that lesson):
if a quadrilateral has two opposite sides parallel and of equal length, then two other opposite sides of the quadrilateral are parallel and of equal length too.
It implies that the straight lines AB and EF are parallel.
This is exactly what we were going to prove.



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