 Hey friends, welcome again to another session on Jim's of geometry and carrying on with our discussion. We are going to deal with some more special type of triangles and their properties. So in this session, we are going to look at medial triangle and its properties. So the first property which we are going to discuss is the medial triangle and the triangle Both have the same centroid. That means both the centroid's coin coincide. So that's interesting, isn't it? So let's first try and see Is it actually happening and if it is happening then how do we Generalize it that is how do we prove it? So in this session will restrict only to Demonstrating that in any given triangle The centroid of the given triangle Coincides with the centroid of the medial triangle of the given triangle It looks a little complicated, but then No problem. Let's first try and you know achieve it. So Here is where I start. So let's say this is first point second point and third point these are the three points three vertices and Now I want to create Triangle out of it. So this is the triangle and Okay, so this is a triangle now my next step is to find out the Midpoints of all these three sides. So let me just try to find that because Medial triangle as you know is nothing but The triangle formed by joining the midpoints of The three sides. So let's say these are the three midpoints Yeah, so you can see three midpoints EF and D these are the three midpoints and now I have to join them to form the medial triangle. Okay, so here is the medial triangle and Yeah, so this DEF is the medial triangle of the triangle ABC. Okay, so first let us try to find out the Centroid of ABC, you know how to find out the centroid So you will just take any two median let them intersect and the point of intersection will be the centroid because we know that third median will anyways pass through the decent, you know the point of intersection of the two Medians so here is where I'm drawing. So this is the first Median second and Third Okay, so let me just name the point. So this point is G centroid now Appearance wise does look like G is also the centroid of DEF, okay, so How do we prove that or how do we even demonstrate it here? So if the medians the given medians are for example, if you take AE is this one this one highlighted one this a sorry, I'll just highlight the Median I'm talking about so let's say this one AE, right? AE is if you see cutting DH DH at you know this point. So if this point happens to be the midpoint of DH then clearly This line here is also the median of Triangle DEF, isn't it? So let's see if this point does particular point is The midpoint or not. So let me try to find the midpoint of DF so if I try so I'm selecting this Segment DF. Let's see where the point comes. So if you see it comes exactly where we were expecting So let me just you know zoom it a bit zoom this in. So I'm sorry Just undo it Undo it. Yeah, so if you see if I zoom Zoom as much as possible if you want if you see this point is the midpoint to which the Median AE of triangle ABC is passing. That means EH is also the median, right? Similarly, if I try to Find the midpoint of EF, let's say so I'm selecting this line EF and see the software has selected this point, which is again, you know, the point through which CD is passing, right? So that means D and whatever is this name I so DI is also the median of DEF, right? So that means G this point G, which is the centroid of ABC also happens to be the centroid of DEF. This is interesting Observation, okay, you can actually prove this very easily, you know, I'm just Giving you in words, but then we'll later on, you know, try to prove it pathetically So we know from midpoint theorem that DE will be this DE here This DE will be parallel to AC and will be half of it Similarly, midpoint theorem says that EF will be parallel to BA and will be half of it. So that means You know if you see D H is parallel to BE and EF is parallel to BD. So B, B, F and E, this is forming a Parallelogram, is it? Parallelogram. Now, if that means if E, BD, E, F, BD, FA, FE is a parallelogram that means BF and DE are the Diagonals and we know that diagonals of parallelogram bisect each other. That means this point M if I Just you know find out the midpoint. Yeah So J, J happens to be the midpoint of both DE as well as BH Correct midpoint of DE and BH. So hence if you see D, J is equal to JE and That means J is the midpoint of DE that goes on to say that FJ is the median of DE, FE. Similarly, you'll see EH is the median and DI is the median, right? And all these medians of DEF lie on the medians of the bigger triangle ABC. That means the point of concurrence of the two sets of three medians is same which is G. That's what this particular session was devoted to. You know show you that the centroid of the triangle ABC as well as its medial triangle are the same.