The problem states "Доказать особину средње линије троугла" which translates to "Prove the property of the midline of a triangle". The midline of a triangle is the line segment connecting the midpoints of two sides of the triangle. The task is to prove properties about the midline, specifically that the midline is parallel to the third side and half its length.

1. Problem Description

The problem states "Доказать особину средње линије троугла" which translates to "Prove the property of the midline of a triangle". The midline of a triangle is the line segment connecting the midpoints of two sides of the triangle. The task is to prove properties about the midline, specifically that the midline is parallel to the third side and half its length.

2. Solution Steps

Let's denote the triangle as $ABC$.

Let $D$ be the midpoint of side $AB$ and $E$ be the midpoint of side $AC$.

Then, $DE$ is the midline of triangle $ABC$.

We want to prove that $DE$ is parallel to $BC$ and $DE = \frac{1}{2}BC$.

Consider triangle $ADE$ and triangle $ABC$.

Since $D$ is the midpoint of $AB$, we have $AD = \frac{1}{2}AB$.

Since $E$ is the midpoint of $AC$, we have $AE = \frac{1}{2}AC$.

Thus, $\frac{AD}{AB} = \frac{AE}{AC} = \frac{1}{2}$.

Also, angle $A$ is common to both triangles $ADE$ and $ABC$.

Therefore, by the Side-Angle-Side (SAS) similarity criterion, triangle $ADE$ is similar to triangle $ABC$.

Since triangle $ADE$ is similar to triangle $ABC$, their corresponding angles are equal.

That is, angle $ADE$ = angle $ABC$ and angle $AED$ = angle $ACB$.

Since these are corresponding angles, it implies that $DE$ is parallel to $BC$.

Also, since the triangles are similar, the ratio of their corresponding sides is constant.

Therefore, $\frac{DE}{BC} = \frac{AD}{AB} = \frac{AE}{AC} = \frac{1}{2}$.

This implies $DE = \frac{1}{2}BC$.

Thus, we have proven that the midline $DE$ is parallel to the third side $BC$ and its length is half the length of the third side.

3. Final Answer

The midline of a triangle is parallel to the third side and its length is half the length of the third side.