Proof that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to half of it:

To prove this statement, we can use the concept of midpoints and the properties of parallel lines.

Given:
Let's consider a triangle ABC, where AB, BC, and AC are the three sides of the triangle.

Construction:
1. Draw a line segment DE, joining the midpoints D and E of sides AB and BC, respectively.

Proof:
To prove that the line segment DE is parallel to side AC, we will use the concept of midpoints.

1. Midpoint Property:
The midpoint of a line segment divides it into two equal parts.

Using this property, we can say that:
- Since D is the midpoint of side AB, AD = DB.
- Similarly, since E is the midpoint of side BC, BE = EC.

2. Using the Concept of Parallel Lines:
To prove that DE is parallel to AC, we will use the concept of parallel lines and their properties.

2.1. Alternate Interior Angles:
When a transversal intersects two parallel lines, alternate interior angles are congruent.

- Consider line DE intersecting lines AB and AC.
- Angle ADE and angle CEB are alternate interior angles as they are on opposite sides of the transversal line DE and between the parallel lines AC and DE.
- By the property of alternate interior angles, angle ADE is congruent to angle CEB.

2.2. Corresponding Angles:
When a transversal intersects two parallel lines, corresponding angles are congruent.

- Consider line DE intersecting lines AB and BC.
- Angle BED and angle BAC are corresponding angles as they are on the same side of the transversal line DE and between the parallel lines DE and AC.
- By the property of corresponding angles, angle BED is congruent to angle BAC.

3. Triangle Congruence:
Using the congruence of triangles, we can prove that triangle ADE is congruent to triangle ACB.

- Side AD is congruent to side DB (by the midpoint property).
- Side BE is congruent to side EC (by the midpoint property).
- Angle ADE is congruent to angle CEB (by alternate interior angles).
- Angle BED is congruent to angle BAC (by corresponding angles).

4. Consequences of Triangle Congruence:
Using the congruence of triangles, we can conclude the following:

- Since triangle ADE is congruent to triangle ACB, their corresponding sides are also congruent.
- DE is congruent to AC (by corresponding sides of congruent triangles).

Hence, we have proved that the line segment DE joining the midpoints of sides AB and BC is parallel to side AC and is equal to half of its length.