 Hello friends, welcome again to another session on gems of geometry and Today we are going to talk about another new concept and this concept is about Pedal triangles, right? So we will be talking about what is a pedal triangle and What a pedal point is and then successive pedal triangles. So let's Start so before we talk about pedal triangles. We have to create one construct one triangle So let's first construct a triangle so ABC is The Triangle, okay, so ABC is the triangle with the vertices a b and c right now we take an arbitrary point P Inside the triangle. Let's say this is let me rename it to be P Okay, so now it is P Now let me drop a pendulars from point P on all the three sides a b BC and C a okay, so right and third one So I dropped the perpendicular on the three sides a b b c c a and now let me Mark these points So let's say this point is D This point is E and this point is okay now when I join these three points D E F Okay, so we get another triangle D E F now this triangle is called the pedal triangle, right? D E F is the pedal triangle. So what is the Characteristic of the pedal triangle so you take an arbitrary point P anywhere inside the Triangle and you drop perpendicular from P on to the three sides then the Triangle formed by joining the foot of the perpendicular in this case D E and F will be called the pedal triangle okay Now there are there would be some special pedal triangles as well. Let's say if point P happens to be the Orthocenter of the Triangle so hence orthocenter is you know Or when P is the orthocenter then also if you join the foot of the perpendicular or the oddities you'll get another pedal triangle similarly when P is the Circumcenter, what is the circumcenter the point of intersection of all the perpendicular bisectors of the sides of a triangle. So if P is the circumcenter Then also if you join the foot of all the three perpendicular bisectors, you will get another pedal triangle, right? Now this point P friends is called a pedal point right pedal point P E D A L pedal point, right? So So you can have infinitely many pedal points from where you can drop up and decolours on to the sides And you will get the pedal triangle Now if you see we can continue this process, right? So now instead of taking triangle ABC if we consider triangle D E F and repeat the process You will get another pedal triangle. Isn't it? Let's try to do that. Let me say, you know, I'm zooming it a little bit now From here, let's do repeat repeat this activity wherein I'm now drop drawing perpendicular On the three sides D E. So this is one Okay, then On E F Right and then finally on F D Okay, so let me just mark the point so that We know which one is the next pedal triangle. So hence, this is the first point G H And I now again, I can join the three vertices To get another period triangle, okay So this is the second pedal triangle. Okay, so the first one is D E F of Triangle ABC. The second one is IGH or GHI Now again, if you see I can go on repeating the process up the image might look a little bit more clumsy But don't worry. All right. So now let me just draw the second pedal triangle Okay, so first second is already done drawn. Sorry. So third one So I have to again drop appendiculars from P on to let's say GH. So this is one from P on to H I so This is the another one is so close. So, you know, don't get confused and then from P on to GI so Here it is. Let me mark the points again. So if you see from P to GH, this is one J Then from here to I think this one, right? Yeah, so this one is the perpendicular K and From P to this one is L. Okay, so let me again Jkl, right? So I'm drawing the polygon Okay, so let me just J K L Okay, so this is if you see this is the Third Piddle triangle guys third Piddle triangle this one You can see that third Piddle triangle, right? Hence you can keep on going like that and you'll get n number of Piddle triangles in the same triangle, right? Now surprisingly, which will be you know, another theorem to be proven is that the the triangle ABC is similar to the third Piddle triangle and in this case the third Piddle triangle is Jkl, okay, so This is one theorem where it says The third Piddle triangle is similar to the original triangle. So ABC you see ABC and C. This is ABC Is similar to third Piddle triangle, which is Jkl We'll prove it in the next session, but for your information, please remember What is the meaning of a Piddle triangle? What is meant by a Piddle point? And you can have multiple Piddle triangles in the same triangle, right? And we say first Piddle triangle second Piddle triangle and likewise nth Piddle triangle, okay? So we'll see some proofs related to Piddle triangle in the subsequent sessions