The line segment connecting the midpoints of two sides of a triangle is called the midline of the triangle. The property theorem of the midline of a triangle is that the midline of a triangle is parallel to the third side of the triangle and equal to half of the third side. According to the judgment that a group of parallelograms with parallel and equal opposite sides are parallelograms, a parallelogram can be obtained by translating line segments.
We can use the translation method when proving the triangle midline theorem. Let D and E be the midpoints of AB and AC on the side of △ABC, respectively, and the intersection point C is the extension triangle of CF‖AD intersection point d e at point F: the line segment connecting the midpoints on both sides of the triangle is called the center line of the triangle. The center line of a triangle is parallel to the third side, and its length is half of the length of the third side, which can be easily obtained by using the properties of similar triangles.
Two inverse theorems are also true, that is, a straight line parallel to the other side through the midpoint of one side of a triangle will bisect the third side; The line segment parallel to one side and half the length of the side inside the triangle must be the center line of the triangle. However, it should be noted that there are two line segments passing through the midpoint of one side of the triangle, which is half the length of the bottom, and not necessarily parallel to the bottom.
Trapezoid: The line segment connecting the midpoints of the two waists of the trapezoid is called the midline of the trapezoid. The center line of the trapezoid is parallel to the upper and lower bottoms, and its length is half of the sum of the lengths of the upper and lower bottoms. The trapezoid can be rotated by 180 to complete a parallelogram, which is easy to prove. Whether the inverse theorem is correct is similar to the above.