Question

If any straight line be drawn from the vertex of a triangle to its opposite side. prove that it will be intersected by the straight line which joints the mid-points of the opposite sides.

Answer 1

Given that, ABC is a triangle

Let D be the mid point of AB and E be the midpoint of AC.

F be the mid point of BC then AF is the straight line that bisects DE at point O.

TO PROVE:- DE bisects AF

PROOF:- since D and E are the mid points of the side AB and AC respectively of $\Delta ABC$, then

$DE\parallel BC$ ………. (1)

since $DE\parallel BC$, then $DO\parallel BF$ [as BF be the part of BC and DO be the

part of DE]

In $\Delta ABF$, then we know that D is the mid point of AC and O be the mid point of AF.

Then, $DO\parallel BF$ (by Converse of mid point theorem)

$\Rightarrow$ $AO=OF$

Hence, $DE$ bisects $AF$.

Hence proved.