okay here we'll prove the hijran's formula and here we go first let's begin with the
triangle and let me just draw it right here and i will call this to be a b and c
and remember the area of the triangle is just going to be one half base times height
and let me take this for the base and i would just say here is going to be the height and
let's say this is b1 and here is b2 now here's the do the area is one half base times height
h if we can write h in terms of a b and c then we will be in good shape so that's the key right
hmm let's see let me just write down the things that we can do from this picture first
for example we know b1 plus b2 has to be just equal to b so let me just write that down and
uh if we look at this triangle this is the right triangle so we can actually just use the path
second one zero for this so let me write down b1 squared plus h squared that's equal to a squared
similarly we can do the same thing right here so b2 squared plus
h squared and that has to be equal to c squared so that's pretty good as well hmm let's see let's go
ahead and just work with these two equations first because as we can see they both have the h squared
so let me just look at this two and then let's subtract the equations when we do that this and
that will be cancelled it and this is going to be b1 square minus b2 squared that's equal to
a squared minus c squared that's pretty good and notice this is the difference of
two squares so of course we can factor it b1 minus b2 times b2 i mean b1 plus b2 like this
and that's equal to a squared minus c squared i am not factoring out right hand side because we
notice the left hand side here this right here is nicely equal to just the b right and uh let's
divide the b on both sides so in another word we can see that b1 minus b2 is nicely equal to
a squared minus c squared all over b and you can see that the right hand side is just in terms of
a b and c which is pretty good and we can make it better because if you look back to this equation
we have b1 plus b2 is equal to b let me just put that down [Music] and i am just going to add the
equations because this way we can actually get rid of the b2 right this and that will be cancelled it
so let's work with this first well let's get a common denominator so this right here should be
over b and i need to multiply by another b so this is b squared plus this and that
let me write yes a squared plus b squared and then minus c squared over b okay now the left-hand side
b1 plus b1 just 2b1 let me divide the 2 on both sides so i get b1 equals this over 2 b like that
yeah so that's what we have for b1 and perhaps we can write it down right here just for some notes
hmm okay i have b1 but i still need to find h what's the connection between b1 h and a well it
was right here right so if you look back here we can subtract the b1 on both sides in another word
h squared equals a squared minus b1 squared and of course we can just take the square root on both
sides of course just do with the positive square root because we are doing geometry so h is equal
to this right here so that's pretty good and i'm going to write that down right here for you guys
as well so h is nicely equal to square root of a squared minus b1 squared that's good and you see
b1 in terms of a b and c i can just put that down right here and that's pretty much the idea right
so let me clean things up right here and i will do the next part so now here is the do the area
of the triangle of course is equal to one half times base times height and of course we have seen
that earlier h is equal to this so we'll just open the square root and inside here we have a square
minus b1 squared and we know b1 is equal to this and that's so nice because b1 is in terms of a b
and c and now we can just write that down and now this formula is actually so so so nice because
the area of a triangle is already in terms of a b and c right however this is not how the
heurons formula looks like so i will do some more algebra for you guys in order to get to
the phone that we want right okay we can combine fractions inside notice on the bottom here is 2b
and then we have to square that meaning that we actually have 4b squared so for this i will
have to multiply the bottom and the top by four b square so they will have the same denominator
so that's pretty good and let me just continue let's have the one over two
b all the way in the front and then the square root both of them will have the
one over four b squared so i will just write that down first one over four b squared and then
notice that for this part is four a squared b squared and a minus this is something square
i think it's a good idea to write this as parenthesis with 2 a b inside and then square
and a minus for this i will keep it a squared plus b squared minus c squared and then squared
because this way we can actually use the difference of two squares to factor out
inside right which is very nice now you notice that we have the 1 over 2 being the front huh
let's invite them inside of the square root house and when you do that be careful though 1 over 2
inside the square root becomes 1 over 4 and then the b on outside inside becomes b square right
so as we can see everything will be inside and 1 over 4 times 1 over 4 that's 1 over 16
b squared and over b squared they cancel out and now we can focus on this factor it out
with the difference of two squares notice here is the first term and this is the second term
now let's go ahead and continue parentheses minus another parentheses that's the first factor and
then another factor is parentheses plus the second parenthesis well what's inside the parenthesis
of course that's just doing everything very nice huh now the question is can we do more of course
notice for this particle if you distribute the negative from here to here you will notice that we
actually get c square first right and then if you distribute this tribute you actually end up with
minus and then we get a squared and then minus b squared right and this right here is a plus
2 a b like so this is the perfect square which is very nice okay
and for this portal if you focus on 2a b and also this well that's also a perfect
square let me write it down first we will kbs a squared plus 2 a b and then plus b squared
and in the end here we have the minus c squared so that's the idea and now if we look back to
this inside we can factor more because here is c squared and i can factor out negative first
if we do that here it becomes a squared minus two a b plus b squared in another word that's a minus
b and a square very nice huh and then here we can of course factor the first three terms is
a plus b squared and then minus c squared this is the difference of two squares and of course this
is also the same thing right so i think now it's a good time for me to write down the square root
again let me not put on the one over 16 yet let me just look at this and then factor it for you guys
right here we will have c minus this and then c plus that right and then for this right here
of course we can do the same thing now here is the time to take care of the 16 and notice that
we happen to have one two three four four factors and 16 is precisely 2 to the fourth power right
so i would write down 16 s this right here over 2 and then over 2 and then right here and then
right here very nice now here we have a plus b plus c over two i'm going to code that to be s
so let me just write that down let a plus b plus c over two to be s and this right here
is what we call the semi parameter because a plus b plus c is the parameter of a triangle
and divided by 2 is of course the semi version of it very nice right
this right here is s how about this though well it's actually really nice as well
if we have a plus b minus c over 2. this right here have a look in fact it's the same as saying
a plus b plus c over 2 and we just have to minus c and you see this right here works because
of course to get a common denominator i need a 2 and a 2 c minus 2 c is precisely the negative c
which is very nice right and the rest can be done with the same way so i'll let you
guys verify that yourself finally i will tell you this right here is s and then next
we have the s minus c and you can imagine we have s minus b and this right here is s minus a so all
we have to do is just write that down so ladies and gentlemen this right here is the heron's
formula for the area of a triangle and this is a proof so i will draw a box and shade this in