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Mid-point theorem: - It says a line segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the length of the third side.

Example: -

In triangle ABC, D is midpoint of AB and E is midpoint of AC.

$

\therefore DF//BC\\

\therefore DF = \dfrac{1}{2}BC $

So we can say that it bisects the line into two equal parts.

Construction: - Draw DG parallel to BF, which meets FC at G.

Proof: - In triangle ABC, AD is median and E is the midpoint of AD.

In and E is the midpoint of AD. Hence, F is midpoint of AG (midpoint theorem)

Or, $AF = FG$ ................equation(1)

In and D is the midpoint of BC.

$\therefore $ G is midpoint of CF (midpoint theorem)

Or, $FG = GC$ .................equation(2)

From equation (1) and (2)

$AF = FG = GC$

Since: - $AC = AF + FG + GC$

$AC = 3AF$

Hence proved.

Converse mid-point theorem: - It states that when a line is drawn through the midpoint of a side of a triangle which is parallel to the second side then it will bisect the third side.

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