 Hello and welcome to this session on Gems of Geometry and in this particular session we are going to deal with another theorem which says that the orthocenter of an acute angled triangle is the incenter of its orthic triangle. Now before we start our discussion on this theorem let's first understand what is meant by orthic triangle. So guys if you have a triangle and you know that there are three altitudes possible so hence if I drop a perpendicular from all the three vertices to the opposite sides so let's say abc is a triangle and I drop this ad as perpendicular to bc and then on this this is let's say be is perpendicular to ac and like that so I'll draw it once again so yes so this is let's say f so hence three perpendicular are there or altitudes are there then if you join the foot of these three perpendiculars so d e f e and f d so we say triangle e f d is an orthic triangle or thick triangle. I hope you understood the concept see the definition of orthic triangle is you drop three perpendiculars in a triangle and join the all the three feet of the perpendicular the resultant triangle will be called orthic triangle. Now let us understand this theorem so the theorem says the orthocenter of an acute angled triangle is the incenter of its orthic triangle okay so in this picture if you can see abc is the given acute angled triangle and e f d is the the orthic triangle now h happens to be the orthocenter where all the three altitudes of triangle abc are meeting so I'm highlighting this h this is the orthocenter here so this is h and we have to prove that h also happens to be the incenter of f d e now what is incenter guys incenter is nothing but point of concurrence so what is incenter let's define that so incenter is nothing but point of point of concurrence concurrence or intersection concurrence of all the three internal angle bisector internal angle bisectors of a triangle right we know this already okay so h has h is also the incenter of e f d that's what we have to prove so let's try and understand this so let's try and understand the give this particular proof so now abc is given as the given triangle and e f d is the orthic triangle so let's try to prove so what we have done is we have done some construction so construction is construction is o is the ortho center sorry o is the circum center o is the circum center of triangle abc so the moment I do that what happens so and o a dash if you see o a dash is perpendicular to bc this is the construction done so since o is the center and a is a point lying on the circles and then hence we can say angle boc is equal to angle a right or rather two times angle a so why because angle subtended angle subtended at center is equal to twice of angle subtended at circumference we know this this is one theorem right so angle boc is twice of angle a and we know that angle bo a dash is equal to bo a dash is equal to c o a dash right right because the perpendicular o a dash will divide angle boc into two equal angles and this must be each of them must be equal to angle a correct right so each of them so hence now let me mention that so this is angle a this is angle a both are angle a clear so if this is angle a guys then what do we get this is 90 degrees this is 90 degrees so can I say this as x and this as sorry this whole as x so you can see x will be equal to 90 degrees minus angle a right by angle some property if you look closely in triangle o b a dash o b a dash so x will be 90 degree and minus angle a by you can write angle some property of a triangle so x is now known so x is 90 minus angle a okay now if this is x so this also is x clear now if this is angle a and this happens to be 90 degrees isn't it e is the foot of the perpendicular so this is 90 minus a this angle angle e b a so I can write angle e b a is also equal to x so this angle is x right similarly in triangle a f c if you see this is definitely 90 degrees so hence this angle here is also x right so angle a c f a c f a c f is also x all of these are x correct so you can identify now which all angles are x now if you see angle h f b or b f h is 90 degree and same is equal to angle b b h b b h both are 90 degrees if you if I draw it these are two 90 degrees right so you can imagine that now you can conclude that b f h d is a cyclic quadrilateral cyclic quadrilateral because in cyclic quadrilateral opposite angles some or opposite angles are supplementary right so b f h d is a cyclic quadrilateral so opposite angles angles are supplementary so hence we can say we can say that b f h d is a cyclic quadrilateral hence I can say angle f b h will be equal to angle f d h f d h and hence this is equal to x why is this it's nothing but angle subtended angle subtended by an arc on the circumference on the circumference are same is that it so if you see if you if you imagine a circle like that something like that then h f is subtending angle b and angle d yeah so hence both of them will be x so hence I am writing this this angle is also x similarly if I say e h dc is a cyclic cyclic quadrilateral again why because if you see clearly this is 90 degree and this again is 90 degree so d angle d and angle e are 90 degrees each so hence what will happen e h dc is a cyclic quadrilateral so you know you can pause and understand this before proceeding so same reason right and both again will be equal to x right oh sorry I have not written the angles as of now so this is now from here I can conclude and this is because of the same reason what opposite angle are supplementary supplementary hence hence this is also cyclic quadrilateral so hence I can say angle e h c e c h which is which happens to be angle you know x is equal to angle e h d e d h rather e d h and both are equal to x so this is x this also is x what is x so let me use this different curve this is x this is also x so if you look very you know look let me zoom out I zoom in sorry so hence if you see there are two two angles which are same right this one so let me use a different color so this angle and this angle are both x now what does this mean this means that h d is internal by sector of e d f zoom out now I hope you understood so hence we can conclude what can we conclude so let me write in this part so that the figure is with us so hence we can conclude h d is internal by sector of angle f d e f d e right similarly and and and what is the value of the angle basically so in this case x is nothing but 90 minus angle a similarly if I have to call let's say these two angles y if this is y then you will we'll conclude that this will also will be y and hence you'll see similarly by similar logic similarly h e h e is the angle by sector of angle f e d or d e f where y you will see will be nothing but 90 minus angle b both are same and similarly h f h f is the is the angle by sector of angle which one so h f is the angle by sector of d f e d f e where if I have to use let's say z for these two z so you'll get z is equal to 90 minus angle c correct so hence we see that h also happens to be point of intersection of all the angle by sector so hence we can say h is in center in center of triangle d e f hence proved hence proved isn't it so what did we prove we proved that e in the ortho center of an acute angle triangle is the in center of its orthic triangle so you can pause this video at times where you do not understand the logic understand it and then proceed okay thank you