 Welcome friends. So we have been doing lots of triangles in these sessions, but never mind You know, you know that this is very important part of your syllabus, but right so hence Let's understand another theorem and this theorem is nothing but converse of basic proportionality theorem now basic proportionality theorem says that if there is a line which is parallel to the third side of a triangle then it divides the other two sides in you know in Proportionately right so that's what the theorem says. So let us say this is a b and I have D and E these are the two points where this line is cutting BC. So by basic proportionality theorem if if D e is parallel to BC then then A D upon D B is equal to A E upon EC that's what we learned, right? This is B P T now converse says that So now I'm going to give you the statement of converse of the theorem of B P T and it says if in a triangle in a triangle a Line or a line segment whichever way you want to write it a line divides divides two sides In the same proportion in the same Proportion then the line is Parallel to the third side to the third Side, okay, so in this case in this case, what is it? It is so let me now write it here. So it's given so given a D by D B is equal to a E by EC and you have to prove to prove What do you need to do you need to prove to prove what? D e is parallel to BC so now you understand why is this converse because in the basic proportionality theorem This was given and you had to arrive at this But now in converse of basic proportionality theorem. This is given and now you have to arrive at this That's why it is called converse isn't it so now Let me start with the proof. So how to prove so in this case, we will use something called method of contradiction method of Contradiction one of the very popular and very effective method of Proving results right method of contradiction. What does method of contradiction mean? It means that you assume that whatever is to be Proven is not correct. That is it is wrong and then with logic you prove that your initial assumption itself was wrong and hence The result so what does it mean? So in this question it is being asked to prove that D e is parallel to BC given a D by DB is equal to a E by EC So what I where will I start from I will say let Let D e Not be parallel be not be parallel to BC this is what we have to prove So hence I am starting with this assumption only that this is not true Let it be not true then somehow by a logic if I prove that this assumption of mine was incorrect Then automatically D e becomes parallel to BC. Isn't it? Isn't it so hence we will say that okay? You are saying that D is parallel to BC. You want to prove that I am saying it's not possible Right, so if it is not possible that means there exists a line Let us say D e dash There exists a line D e dash therefore I can write there exists There exists a line D e dash such that such that e dash is on AC and D e dash is parallel to BC Right, so when you're saying that okay, you know D is not parallel then there must exist one line D e dash let us say let us I'm assuming that that line is D e dash such that e dash is on AC and such that D e dash is parallel to BC right? because if you're saying that D is not a parallel line to BC then There must be one line which is passing through D and Which must be parallel to BC there exists a line there is a line BC It must have another line parallel to it which passes through D and let that be D e dash Now if D e dash is parallel to BC then What is the logic? By our BPT now BPT we have already proven and I can use this theorem and say if D e dash is parallel to BC in the triangle ABC then ad by db is equal to a e dash by e dash C Isn't it? This is what? We will get by BPT and now let us say this is this relation was one and This relation by was two so you see the left-hand side of one and two are same So very comfortably I can equate the right-hand sides so I can say a e by e c is equal to a e dash by e dash c right by equating the right-hand side of one and two adding one to both sides you will get a e by e c plus one is equal to a e dash plus e dash c Or other I'm sorry. This has to be step by step. So let me write a e by e dash by e dash c plus one So I have added a plus one to both sides now And this will give you a e plus e c if you take LCM and this is e c This is equal to This is equal to a e dash Plus e dash c divided by e dash C Isn't it now look carefully a e plus e c in the figure Here is the figure. Where is the figure? Yeah, a e. Yeah, so a e plus e c will give you a c isn't it and a e dash and e dash c will also give you a c So hence if you see a e plus e c is a c Now this was divided by e c is equal to a c again because both summed Will give you a c You can check once again if you're not convinced a e dash by a e dash plus e dash c is a c and a e plus e c is again AC right so hence if you see I can strike off a c from both sides this would mean E c is equal to e dash C is it it now? How is that possible that there are two line there are two points e and e dash to a distinct points as well and Are lying on the same line and are collinear so e e dash c are collinear and Right this relationship is also holding this will happen only when only possible only when Only when e and e dash coincide Coincide e and e dash e and e dash Coincide isn't it that means if e and e side coincide e and e dash coincide That means de dash so de dash and De are same Lines That means now we know that de dash was parallel to be see so hence and Since since de dash was parallel to BC therefore Be is parallel to BC so hence proved hence Proved that's what we wanted to prove that de is parallel to the third side if ad by db is equal to a e by Easy, this is the inverse of Basic professionality theorem now we learned lots of corollaries to be PT in the last session so if any of these corollaries are also true then De would be parallel to BC. What do I mean? I mean that if you have a D upon a b is Equal to a e upon a c then also if this is given. Let us say then also De then also Then also de will be parallel to BC Understood. This is called converse of basic proportionality theorem