 Hello friends, I welcome you all to another session on gems of geometry and Continuing with the previous session. We were discussing properties of X circles and in circles of a triangle So in this session also we will be discussing some interesting facts about X circles and in circles So, you know what an X circle is and you know There are three X circles as you can see in the diagram So the X circles are nothing but circles which are tangent to Extended sides of the triangle two extended sides of the triangle and one side of the triangles So if you see these are the extended sides of the triangle and this one is the third side and This circle here this one is the X circle one of the X circles Okay, similarly there are this one is another X circle this one is another X circle and this one is the in circle now in this session We are going to discuss few more properties and first is the in circle and the three X circles are also called for try tangent Tri-tangent circles of the Circles of the given triangle of the Triangles right. So what all so try tangent circles are nothing but three X circles and One in circle Okay, so always remember this definition now if you see and we also know that tangent From the same point on to a circle are equal. So hence by that logic if you see Let's take point B here point B this one So hence we will get B X a will be equal to B Z C isn't it? why because Tangents tangents and gents drawn Tangents drawn on a circle on a circle on a circle from The same point Okay, they are equal Same point are equal Okay, so B X a is equal to B Z B Similarly, we can say with the same logic B X B is equal to B Z B isn't it this is that be here and this is X be here and B was here You already saw that so B X B is equal to B Z B And now can I write B X B plus B Z B? Let me add these two what do I get so B X B if you see closely it is nothing but BC plus C X B isn't it and then the other B Z B is nothing but B a plus A Z B correct Right, so that means can I not write this as BC plus C Y B C Y B why because if you see C X B is equal to C Y B Okay, Y B is here and now B a let it be as it is and A Z B can be written as a Y B a Y B Isn't it now if you look closely C Y B and a Y B added together is nothing but a C Isn't it so hence I can write BC plus a C plus B correct or a B plus BC plus C a Which is nothing but two times semi-parameter s Right, so this is the first learning so what is the learning and learning is Some of any two tangents, you know, so for example B X B two plus B Z B is nothing but twice the Semi-parameter, okay now let me do the further analysis here What next if you see? a y a let's consider a y a where is a y a just check a y a so a Y a so y a is Here, this is why a and this one is a so a y a will be equal to clearly a Z a same logic tangent from the same point and this will be equal to B Z B B Z B and this will be equal to B X B because all of them if you see will be equal to C X C is equal to C Y C and all of them will be equal to Yes Isn't it Right, why because if this to B X B plus B Z B was 2s and we knew that B X a is equal to be Sorry B X B is equal to B Z B then hence you can you can say from here You can say B X B into 2 is equal to 2s This implies B X B is equal to B Z B is equal to S and likewise all the other pair of External tangents will also be equal. So hence you will get this result. So remember this all the you know Tangents are equal to semi-parameter, right now Also, we know that C X B if you check C X B, where is C? This is C and here is X B. So C X B can be written as B X B Minus BC is it at minus BC? So which is nothing but B X B We know it is S and BC can be considered as a so S minus a and if you do the same thing If you do the same thing, you know for all others you will get the same Result, what is that? So B X C now, let's say B X C B X C if you see again, it is nothing but C X C C X C minus BC which is equal to C X C, you know, all of them are S. So S minus a you'll get S minus again Similarly, you can write C Y a is equal to C Y a is equal to C X a Is equal to a Y C all of them will now become equal to a Z C is equal to S minus B similarly the same logic you can prove this as well and finally a Z B is equal to a Y B is equal to B Z a is equal to B X a Is equal to S minus C? So if you see all of them are related to semi perimeter and the length of one of the sides, correct? So spend some time, you know pause the video and you can always understand these relationships So if you understood these two, I just rewrote the result. That's it. Okay, so you can try and arrive at the same result. So let's See some more properties of these type of triangles in the next session