 Now this theorem is the converse of the previous theorem what we saw okay, so What is this theorem now? So it says if two chords a b and BC of a circle With center o so you can see this is the center o here So center o is here and a b and b c are the two chords such that the center o lies on the bisector of angle a b c So what do we know? Angle alpha is equal to angle beta. So alpha is equal to beta. This is given then the chords are equal. We have to prove that a b We have to prove that a b is equal to b c This is what we have to prove. Okay, so How should we prove it? So a b is equal to b c so to prove two sides equal You know that if you somehow prove that the two, you know triangles which are you know seen over here are congruent Then it looks like you can prove that the corresponding sides are equal So hence we have first joined o a and o c to complete the triangles Okay, so let's try the conventional method how to prove so hence we will be writing first of all given What is given guys a b and b c? b c r to two chords Okay off circle With center o and let's just radius r. So these Radii are though. We don't need it, but then for our You know understanding. Let's write that that ways. Okay, you have to prove What do you need to prove guys to prove besides that it's also given so this was given a given b What is given b so? angle a Bo is equal to angle C Bo right because Bo is the angle bisector now to prove a b is equal to bc Okay, and a construction. What do we need to do? We simply need to join o a and o c are Joint Fair enough. So this is Now we have two triangles and We need to somehow prove that they're congruent and then our job will be done So to prove that these two triangles which two triangles. I'm anyways talking about so I'm talking about this this one and That one Okay, so the two triangles o a b and o c b So one angle is definitely, you know common so a alpha is equal to beta We know and o b is the common side But then we have o a is equal to o c because they are the same radii But then there is no criteria like s s a or a s s if you see here We are getting a s s criteria isn't it angle one angle is Equal one this side is also equal Bo and other side is equal, but then there is no such Criteria a ss. Okay, so what we do So we have to get to you know, we have to use the use some other information to you know Get to some Conventional criteria. Okay. So what do we do proof? That's if you see in Triangle o a b Oh a b o a is equal to o b is equal to r O a is equal to o b is equal to r right therefore angle o a b is equal to alpha Isn't it opposite angles Angles opposite to To equal sides equal sides Sides in a triangle are equal in triangle are Equal this we know. Yep. So o a b is alpha similarly Similarly the friend word similarly Reduces so much of effort. So we can say o Cb o cb is equal to beta Right, but we know that alpha is equal to beta. It's given Given isn't it therefore we will have o a b is equal to o c b angles Correct. Now we have got the ingredients. Oh, let's say we'll write write it as gamma So hence Here is the angle gamma. Here is the angle gamma Now let's see what to do. So now you see in triangle Oh a b o a b and Triangle o Cb very clearly Oh a So two angles are common anyways, so we can see Angle a b o is equal to angle Cb o is equal to Alpha or beta whichever you want to say then Angle o a b o a b is equal to angle Ocb o cb which is gamma. We just proved and O b is equal to ob common side common side Right, therefore Therefore by a a s criteria right angle angle side two angles and side criteria We can say that triangle a b o is congruent to triangle Cb o a b o and Cb o order of the vertices mind you that you please mind that a b o and Cb o right therefore What can we say we can say a b is equal to? Cb why is this Congruent part or corresponding parts of congruent triangles right and hence proved so hence we just proved that we proved that if The center lies on the angle by sector of two chords angle made by two chords then the two chords are equal right Hope you understood this theorem and the proof