What is the inverse theorem of midline?

The midline theorem is that the midline of a triangle is parallel to the third side (not in contact with the midline) and equal to half of the third side. There are two inverse theorems:

1. Within a triangle, the line segment that intersects both sides of the triangle and is parallel and equal to half of the third side of the triangle is the center line of the triangle.

2. Within a triangle, the line segment passing through the midpoint of one side of the triangle and parallel to the other side is the center line of the triangle.

Prove:

It is known that in △ABC, D and E are the midpoint of AB and AC, respectively. Prove that DE is parallel to BC and equal to BC/2.

The parallel lines passing through C and AB intersect the extension line of DE at G point.

∫CG∨AD .

∴∠A=∠ACG。

∠∠AED =∠CEG, AE=CE, ∠A=∠ACG (with braces).

∴△ADE≌△CGE .

∴AD=CG (the corresponding sides of congruent triangles are equal).

∫D is the midpoint of AB.

∴AD=BD。

∴BD=CG。

And ∵BD∨CG

∴BCGD is a parallelogram (a set of parallelograms with parallel and equal opposite sides).

∴DG∥BC and DG=BC.

∴DE=DG/2=BC/2。

∴ The midline theorem of triangle holds.