 Hello friends, welcome again to another session on gems of geometry and In the last two sessions, we saw what our medial triangle is what the boiler line is and In the previous session we also saw that the centroid of the triangle and its medial triangle both of them are coincident Okay, and now in this session. We are going to talk about another very interesting fact and that is the orthocenter of the Medial triangle and the circum center of the triangle You know They are same right so hence what did I tell you that the circum center of a triangle and The orthocenter of the medial triangle Are the same that means they are coincident So let's try and construct the triangle and see for ourselves whether it is really true and let's try To prove the same. Yeah, so let's begin with drawing a triangle So here we start. So let's say we are now trying to draw a triangle So let's say this is my triangle ABC and As you know, we can draw the medial triangle by first finding the midpoints of all the three Sites and then joining them. So this is the this is Rather I should first try with the BC. So D Then E and then finally F. So D E F. These are the points which are Just a minute. I'll accept. Yeah, so B is the side length and E is The midpoint of AC. So let's draw the Let's draw the Medial triangle. So now let's try and draw the medial triangle for that I need to just join them join the three midpoints. So D And from E to F and from F to Right. So this is a medial triangle guys. Now what? So as I said, we have to first find out the Circum center of the given triangle. So circum center as you know is the point of intersection of all the perpendicular bisectors of The triangle sides, right? So let me draw the perpendicular bisector Okay So this is the perpendicular bisector of AC, right? Then let me also draw perpendicular bisector of BC. So this one and Perpendicular bisector of this. So if you can see, let me that let me just zoom in a bit Okay, so Yeah, so this is the so let us now first try to understand where is This perpendicular, you know, this is the circum center of our Triangle ABC, isn't it? This is a circum center of triangle ABC. G is the circum center of triangle ABC clearly now if you see This line, which I'm going to highlight now this one is is Perpendicular to AC is it because it is a perpendicular bisector and From the midpoint theorem in a triangle, we know that if we join two midpoints of two sides of a triangle Then it automatically is the it is it is back parallel to the third side. That is fd is Parallel to AC by midpoint theorem so my friends if This line I this one, which I have highlighted is perpendicular to AC Then it has to be perpendicular to fd as well, right? That means this angle has to be 90 degrees Which says or which indicates that let me just name this point. So let's say this point is H So each my friend is the altitude of triangle e fd. Isn't it? Right similarly with the same logic if you see d and let's say this point here is I capital I so di happens to be the altitude of the triangle f e d from vertex d is it and similarly this point if you see J so fj is also the altitude of triangle E fd that means the altitudes of triangle the medial triangle and the perpendicular bisectors of the bigger triangle ABC are Actually, you know collinear so correspondingly so hence, let's say the this particular altitude e h of e fd Lies on the perpendicular bisector I This I small i of AC. Isn't it likewise? Dg right Dg is the altitude Sorry di is the altitude and it lies on j small j, right and Fj is the altitude of the smaller medial triangle Which is coincident or collinear rather collinear with or coincident is a better choice of word On the perpendicular bisector of side BA of the bigger triangle That means if they are they are all one-on-one, you know coincident then that means their point of intersection also will be same So hence g this point g which happens to be the circum center of ABC also becomes ortho center of F e d or a D e f so that means we can now you know No beyond out that the circum center of any triangle and the ortho center of the medial triangle of the given triangle Are coincident? Okay