 Hello friends welcome again to another session on triangles We have another theorem here, and we will try to prove this theorem The theorem says the line drawn from the midpoint of one side of a triangle Parallel to another side by six the third side now This is you have already learned as a converse of midpoint theorem, which we studied maybe in the previous grade So Let's try and first draw a figure for it. So here is a triangle I'm drawing a triangle a bc. Okay, so here's a triangle a b c Okay, and we have to find out or let's say first. This is point D, which is the midpoint Okay, and what I'm going to do is I'm going to draw a line parallel to the third side Okay, let's say it cuts a c at e Okay, so let's write given first. So what is given? D is midpoint of midpoint of a b and What is what else is given D e is parallel to bc. Okay, what is to prove? We have to prove What do we need to prove that D e sorry you have to prove that you have to prove that e is the midpoint Midpoint of Ac then we can say that this line D e bisects the side ac, isn't it? So let's try and prove this how to go about the proof. So let's try and prove this theorem So clearly there are parallel lines given So the first thing which comes in mind is bpt in a triangle basic proportionality theorem And also given that D is midpoint since so we write since D is the D is the midpoint midpoint of Pitchline a b correct right therefore This means what ad by db is equal to one one is two one or one whichever way you want to write correct now since D e is Parallel to BC So a line is cutting two sides of a triangle, which is parallel to the third side then by basic proportionality theorem or Thales theorem basic proportionality Proportionality theorem We know That what do we know basic proportionality theorem says if a line divides two sides of a triangle then a line parallel to the third side divides two sides of the triangle then the The ratio of the divided sides is same Converse is also true That if the ratio is same then the you know the line which is dividing the segments in doubt that ratio is parallel to the third side So by basic proportionality theorem, what can we say or we can say that ad by db Is equal to a e by ec right a e by ec But from one so let's say this is one from one and two So if you see from both one and two LHS is same therefore from One and two What can we say we can say a e by ec is equal to one or a e is equal to ec in that case only then it is one the ratio is one right so a e is equal to ec Let's check the figure now. So the moment a e is equal to ec. What will this Conclude this will be you know, this will lead to the result that e is the midpoint of a c or Be bisects a c Isn't it Understood. So this is how using basic proportionality theorem. We prove that If a line starts from the midpoint of one of the sides of the triangle and is parallel to the third side Then it will bisect the opposite side as well, right? So remember this theorem