 Hey, hi friends welcome to another session on gyms of geometry and As discussed in the previous session in this session, we are going to deal with few more properties of the pedal triangle and We'll start with this theorem which says that if x y and z are the distances of a pedal point P from the three vertices a B and C of a triangle ABC Then the pedal triangle has sides a x upon 2r b y upon 2r and Cz upon 2r Okay, so in the last class we studied what is a pedal triangle? So you take any point P like this here so if you take point P Like that and then you drop perpendicular on all the Three sides so you can see PB P. Sorry PD P E and P F are the perpendicular drawn from P on the three sides Okay, a P is given to be equal to x. You can see this is x here a P is x B P is y. So this is why and Cp is z Okay, so this is what is given now we have to prove that we have to prove that The three sides of the pedal triangle. What is the pedal triangle if you join the foot of the perpendicular? Drawn from P. You'll get a pedal triangle in this case. We have a pedal triangle D E F right now you have to find the length of the sides D E F and F D So let's see how do we find it now if you see triangle or I'm saying angle a a F is equal to angle a EP both are 90 degrees right, so we can say very clearly that a P is The dia is the diameter diameter of a Circle of a circle Um on which on which F and E Like correct why because we know that if there is a circle and there's a diameter so the diameter subtends 90 degrees on the circumference is it so this is what we have known Okay, now what so if that is the case that means a F E So the circle so AP is the dia of the circum Circle of triangle AFP so we can conclude that a P is the diameter AP is the diameter and what is the circle which is you know circum circle of diameter of the Circum circle circum circle of triangle triangle a E F Isn't it circum circle of triangle AEF This is what we conclude so AP is the diameter of the circum circle of AEF now If you remember sign rule we which was discussed in the very first session which says that sign a By or rather the other way round. Let me write in the reciprocal format. So we say that F E in triangle AEF F E by sign of a Will be equal to two times the Radius of the circum center, right? So two times the radius of the circum center in this case Let's say R. I am writing and in the subscript. I am writing AFE just to indicate that our AFE is the Circum radius of triangle AFE, which is nothing but AP because this is the diameter of that same is for that circle. So hence we can say F E by sine a is AP so hence we can conclude that sine a is equal to F E By AP and AP is nothing but X Okay, similarly, I can say in triangle a BC What can I say in triangle ABC BC upon? BC upon sine a is equal to two R where R is the circum radius of triangle ABC R is the Circum radius Circum radius of triangle ABC Right guys. So if you see the nomenclature here would be this is side a This is side B and this is side C, right? So hence I can now replace BC by simply a so a upon sine a Is equal to To R Right. So hence from here I can conclude this implies sine a is equal to a upon two R Now let this be equation number one and this be equation number two So if we can if you just you know unite both of them, you'll see F E by X F E by X is equal to a upon two R So this implies F E is equal to a X upon two R So we got the first side of the pedal triangle D E F and which is nothing but F E right F E Is equal to a X by two R? similarly, it will not take much of Similarly, you can always prove that F D the other side will be BY by two R and E D will be CZ by two R By the same logic, right? So if you understood this Particular logic, it will not be difficult for you to come up with this conclusion, right? So we just prove that if XYZ are the distances of a pedal point P from the three vertices A, B and C of Triangle ABC then the pedal triangle has these three sides Okay, and you can check the one special case would be when So let me just write it here. So special case one special case would be when X is equal to R that means The point P itself is the circum center E is the circum center then X equals to R and then what will happen? You know all you know very clearly so the sides of the The three the three sides of the pedal triangle in this case will be simply A by two B by two and C by two half of these sides, right? We know that the you know if you join the midpoints of the Sides of the triangle the Triangle so formed will have these three sides in this right so that the special case of this particular theorem Okay, I hope you understand this theorem