 So, we just saw that the theorem which is written over here works through the GeoGibra demonstration. We could show that if A, B, and B, C are two equal chords of a circle, then the center of the circle lies on the angle bisector of the angle A, B, C. We saw that through a demonstration on a GeoGibra platform. Now we are going to prove this theorem. And how do we prove this theorem is like this. First, let's write, you know, the conventional proof methodology is what's given. So, the given part is that A, B is equal to B, C, R chords of circle, let's say, capital O small r. What does it mean? So, O is the center, R is the radius. This is how the circles are depicted. Now, we have to prove, to prove, to prove. What do we need to prove? Center O, center O lies on the angle bisector, angle bisector of angle A, B, C. So, for that, we need to construct something as well. So, what do we need to do? So, here is the construction part, construction. So, we have to construct something here. So, we say that we have to join A, C, A, C joined, A, C joined, okay? And A, D, the bisector of angle A, B, C, drawn, right? So, this is the construction. So, if you can see in the diagram, not A, D rather. B, D, B, D, sorry, my bad. So, let me just make it more clear. So, B, D. So, B, D bisector of angle A, B, C, drawn, right? Now, let's try to prove. So, here is the proof. What is the proof? In triangle A, B, D, A, B, D, and triangle C, B, D, C, B, D. If you see, angle A, B is equal to B, C, given, equal chords. So, given, and alpha is equal to beta, right? Because B, D bisects, bisects angle A, B, C, isn't it? And third is B, D is equal to B, D, common. Common side, isn't it? If that is true, then let me write it here. Then we can say that triangle A, B, D, A, B, D is congruent to triangle C, B, D. C, B, D, isn't it? By which congruence law? By SAS criteria, isn't it? Side angle, side criteria, we prove that these two triangles are congruent. Now therefore, I can say angle B, D, A, B, D, A is equal to angle B, D, C, okay? And the reason being equal parts of congruent triangles, corresponding parts of congruent triangles, C, P, C, D. Also, I can say A, D is equal to D, C. Why? Same reason, C, P, C, D, okay? That means what? This is the midpoint. So D is the midpoint, midpoint of AC. Also angle B, D, A plus angle C, D, A, or B, D, C, rather, or C, D, B is equal to 180 degree. Why? Linear pair. We studied linear pair on one line, the total angle is 180 degree at a point. So therefore, I can write twice angle B, D, A is equal to 180 degrees. Why twice angle B, D, A? Because B, D, A, and C, D, V, or B, D, C are same, where it is written here. So hence, angle B, D, A is 90 degrees, right? Therefore, D is the midpoint and B, D, A is 90 degrees. What does it mean, guys? It means that B, D is the perpendicular bisector, perpendicular, so this symbol I'm writing is perpendicular, perpendicular bisector, perpendicular bisector of AC, AC. Now guys, AC happens to be a chord, right? Therefore, therefore, and AC happens to be a chord and B, D is the perpendicular bisector of the chord, but we know that perpendicular bisector of the chord always carry the center. So hence, we can say that B, D will carry or contain the center. And why is that? Why is that? Because perpendicular, so perpendicular bisector of a chord passes through the center, passes through the center and hence proved, hence proved, right? So what did we learn? We learned, first of all, we demonstrated that this theorem actually works, it's valid and then we worked through its proof, right? So this theorem is proved.