 Okay, so in this session, we are going to discuss another important topic and that's called 9-point circle in a triangle So what we are going to do is we are going to draw a triangle and identify or you know Construct a 9-point circle and Then try to see its properties and in the later session will also try to prove few properties related to 9-point circle So let's first start with what exactly is a 9-point circle. So in the previous sessions, you studied about Medial circle, medial triangle and what was medial triangle basically So the triangle formed by joining the midpoints of the sides of a triangle and then we also learned concepts of ortho-center centroid In center, circum-centers and things like that. So we will be using you know those concepts again and It's some special You know attributes added to them. We'll see how 9-point circle is generated. So basically The 9-point circle is nothing but as the name suggests there would be nine points We have to identify those nine points. So those nine points are nothing but the foot of the Artitudes in the triangle the midpoint of the sides of the triangle and The midpoint of the segment joining the ortho-center and the vertices. So instead of explaining in words, let's try and explain Would you have drawn? Okay, so let's first draw a triangle. So This is a b Let's see and a b c is the triangle now We will try to find out what what did I say these as we are discussing nine-point circle. So we must have Those nine points. What are those nine points first three points are nothing but the midpoint of the even sides so a b First draw this. Yeah, so a b Bc is D and then this is e And So these are the midpoints of the Sites of a triangle right now. What do we do? We have to draw? Artitudes right so let's draw altitudes here. So first altitude is This one and then second is this one Yeah, second one is From this side to So this is second altitude and third one as you know will pass through the point of intersection But this altitude intersection point is too close to be so let me just change the position of b so that it becomes more clear, yeah So this looks good, right? Now what we are going to do is we are going to find out and first before that Let's first name the point. So let's say this point is G then H and I and J like that point of intersection has to be there. So this point and Right So now these are how many points did we get so let's count them So first of all D E F three points and I it's I J six points. We need three more. What are these three more? nothing but midpoints of Midpoint of BG AG and GC. Okay, so midpoint of BG. So let me draw Yeah, so K is the point and secondly of AG. So L is the midpoint and third C G so M is the point. So did you see there are nine points now what all? Let's start D H K F J L I E M right, these are the nine points and now to our surprise if we try to join all of them that's right If you can see we get a circle passing through all of them. So that means all these nine points Lie on the same circle. Isn't it interesting guys? So All of them lie in the same circle now You'd be have you'd be thinking that maybe this is because one particular configuration of the triangle So to eliminate that doubt, let's try to you know, see different different triangles. So let me just reposition B So as I'm changing B, you can see in every such triangle. So these are all random triangles and you see you always get a Circle this particular circle is called Nine point circle, right? So let's say now. This is an equilateral triangle almost and Yes, so you can see you you will get yeah, but now what happens is the triangle disappears The moment it becomes an obtuse angle triangle, but is that so so let's first try to make an obtuse angle triangle So if you see now, I have made a obtuse angle triangle obtuse angle triangle So angle ABC is an obtuse angle and The orthocenter G in this case is lying outside the triangle has to be but let's see you in this case also We get a circle. So hence again if we join all those same nine points, okay? So you can see again You're getting a circle. So let me now try to move this B So hence whether the orthocenter G is inside circle or sorry inside the triangle or it is outside the triangle You will always get a nine point circle. Isn't it interesting? So let me just move this point a right. So if you see as a is moving Irrespective of that fact, we are getting a nine point circle all the time and let me make this now obtuse So, yeah, even then You can see there is a nine point circle Hi Hi, so I hope this is clear to you and similarly if I move C also you'll see So this is the concept of nine point circle now There are lots of properties attached to this nine point circle which will take up in the successive sessions