Here are two classic geometry triangle and star polygon proofs that might be really hard at first! Do you think geometry is hard? Especially the geometry problems in the math olympiad or competitions? Well, I certainly think so and I will tell you why!
do you think geometry is hard i personally think that geometry is really hard especially the map
context geometry questions because sometimes when you're trying to solve a question or proof
of theorem you have to draw the parallel line or maybe that perpendicular line or maybe from
point a to point b well maybe draw a happy face in order to solve the question right
man how in the world do we know to do that well today i would like to give you guys two examples
to illustrate what i just said and i'll tell you guys more about my feeling about geometry
all right so for the first one let's take a look at the triangle right here and perhaps the most
famous fact about the triangle is that the angle sum inside is 180 degrees so let's put on a b
and c like so but how can we prove that though well if you have never done this before then
you might not even know how to start it but once you see it you see it this is almost like looking
at the 3d picture right once you see it you see it the way to do it is you pick a side let's say
we pick this side and we go to the other vertex and just go ahead and draw a parallel line like
so well once we have this parallel line and this parallel line is achievable by the euclid's fifth
postulate right well what next thanks to this parallel line we know that this angle
is the same as that angle so this will be angle measure a as well and likewise this angle will
be the same as that angle which is going to be c and now have a look do we know a plus b plus c now
a plus b plus c we don't know maybe but we do know once we look at a plus b plus e because
they form a straight line so we know that has to be 180 degrees and guess what we are done
thanks to that parallel line and that's exactly what i was saying like how in the
world did we know to draw that line right okay let me take a look at another example
do you know is there anything else that has angle measured out to be 180 yes a star so let's take
a look at the star so let me show you guys a nice thought let me put down a star like this
like this right so let's look at the vertex thesis right here so let's say a b
c d and e we wanted to show that a plus b plus c plus d plus e add up to be 180 degrees as well
but how can we do that well check this out this time what you can do is just pick any vertices
like this and that this and that but whatever let me just go ahead and do this and that right and
i'm actually going to consider these two angles as well so i wanted to put down little f and detail g
right here okay what's next have a look well i'm going to be focusing on this triangle right here
so let me color this in red and also we will be looking at this triangle right here
have a look you see that this angle must be the same as that angle because there are vertex
angles right good and then what's next though well these two angles must add up to these two angles
why because we know that these two angles are already equal so we must know that b plus d
has to be equal to f plus g and notice i'm not saying f is equal to b where f is equal to d
we don't know we're just saying that the sum of this 2 must be equal to some of these two
and notice that i purposely drew this line this side like this way because this time we are not
drawing a parallel line right but anyway now let's take a look at the original question which is a
plus b plus c plus d plus e okay we know that b plus d is equal to f plus g then let me put down
a plus c plus e first i want to write this down first because if i put down f right here then it
looks like b is equal to f which is not y one so let me just put this two at the end like so great
excellent now i'm going to consider this triangle so you just have to be really observative
look at the picture from like one little part to the big picture and so on so on
have a look why do we care about this triangle yeah we have a plus c plus e a c e and uh
f and g that's a triangle and we did that over there already well we must know that this right
here add up it will be 180 degrees so the sum of all these angles at the vertices of the star
is also equal to 180 degrees which is really cool but again the question is that how do we know to
connect the vertices right here and then consider these two angles right
i think geometry requires a lot of experience and you might not see a lot of geometry videos on my
channel because i personally do not have that many experiences to be honest with you guys and also
they are hard yeah and one of the ways to get better is that just of course practice practice
practice and when you are encountering new questions like you see we drew this kind of lines
if you have enough experience then maybe when you are encountering new questions you will be like oh
maybe i will try that i'll draw the parallel line or maybe oh i will draw the perpendicular line
right so that's the way to go you can check out my other video where i draw perpendicular lines
to prove the law of sine law of cosine and things like that over there right and as always that's it