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hi everyone welcome back uh in today’s episode we’re going to cover every point

is incident with at least one line this is seems like an obvious fact

but remember we’re doing incidence geometry

and so let’s do some math let’s go ahead and get started on this so we’re going

to recap what incident axioms are so we’re talking about point line and

incidence as being undefined terms and we’re going to have three axioms so

the first one is for every point p and every point uh not equal to p

q let’s take another point q there’s going to be one unique line

incident that passes through or incident with p and q um and so the

point of these three actions is to give meaning to points lines and incidents

so axiom two says for every line just draw any line you want

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and there exists two points on it at least two points and axiom a3 says

there exists three uh distinct points with the property that no line is

incident with all three of them um and then we have some basic

definitions that we’ve been talking about by the way if this is going too

fast for you this is a recap the link is in the description below

uh that starts uh uh the whole series yeah so you want to check out the series

um so we have the the definition of collinear and concurrent lines

and we have the definition of parallel so we haven’t talked anything about

parallel yet but we’ve been talking about collinear and concurrent and we’ve

been using these axioms to prove uh different statements so what’s been

proven so far in a previous episode we proved um

theorem one here and in the previous episode we proved theorem two there

exists three distinct lines that are non-concurrent

and in this episode we’re going to prove

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that every point is on at least one line so whenever we’re proving theorem three

here the idea is that we can use any of our three axioms

and either of these two theorems because we’ve already established them

and then in upcoming episodes we’re going to prove these other statements

here check out the next coming episode soon

all right so let’s start off with the proof here every point is incident with

at least one line now that seems pretty uh obvious for example if i just put a

point down and i can say oh just draw a line through it right so

but how do you know you can do any of that right and so that’s we want to

justify using our incident axioms all right and so we’re going to write a

proof out here and and when we write this proof here

i’m going to go through this step by step in a column format and then when

i’m done i’m going to put it in a paragraph format and then

and then we’ll talk about a couple of other things all right so we’ll talk

about a statement here and i’m going to number these also so my first statement

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is going to be that p is a point so i’m talking about every

point right so i just want to give a name so i’m going to say p is a point

and i’m not saying there exists a point or anything like that i’m just saying

that this is my hypothesis is that i i have a point and i want to show a line

goes through that point so i’m not really subscribing any kind of

properties or or saying anything specific about p i’m just saying it’s a

point and and and our justification is uh this is our hypothesis

every point right so i pick one so every point so p is a point

and i want to show that there’s a line going through p

that’s incident with p p is incident with some line

so how do we do that well we have an ac step one so we have an axiom that says

that gives us three distinct points and i’m going to do i’m going to use that so

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i’m going to say a b and c are distinct non-collinear points the non-cleaner

part we we’re not going to need in this but i’m just going to say that anyways

our a b and c are distinct non-linear points and so this is by axiom a3 here

step two step three um is well i have three

points here none of these are equal to each other now this point p

now here’s the thing that a beginner might be stuck with is well i

named it p can’t i just name it a or b or c um or can’t i name one of these p

so these are just names you can’t give any special meaning to them so if

you put a here then you’re giving it meaning that we

got the same two points in these two steps here which we don’t necessarily do

we don’t necessarily know this is every point right so we cannot

we cannot say anything special about this p p is any point

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and here we have a b and c are existing because of axiom a3

and we don’t know anything about these a b and c we don’t know anything about

these three points so p might be one of these three points or it may not be one

of these three points we just don’t know and so because of that we’re going to

use a point of logic we’re going to say exactly one of these holes

exactly one holds and that’s that p is equal to a

or p is not equal to a and so here i’m picking on a

uh we could have written step three down

for p is equal to b or p is not equal to b or you could have picked on c if you

wanted to but this is certainly a point of logic p is either equal to a or p is

not equal to a and so i’m just going to call this the law of excluded middle

law of excluded middle and that’s just a

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tautology from logic there that’s just a point of logic

so now what this means is that i have two cases

that p is either equal to a or it’s not and so i’m going to go with this is my

case one p is equal to a so now what we have here is because

these are three distinct points i know that axiom three says a is not b and b

is not c and c is not a i know these are different points here

so if a is if p is equal to this one then i certainly know that p is not

equal to this one so i’ll say p is not equal to b

and then i’ll just say this is true by uh step two

and so what we have here now is two distinct points

and so now we can say there exists a line that passes through p

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there exists a line l incident with p which is what we’re trying to show there

exists a line that’s incident with p and this is by axiom a1

there’s axiom a1 there and i know that i can use axiom wave one

because i need two distinct points and i have two distinct points i have a and i

have b and a is in fact p so there’s a line that goes through p and b

um and now we can come up here and start numbering them again so i’ll just say

step 7 here now this step six this case right here

case one is done because i have exactly what i want i have a line passing

through p well we have the same thing in case two

so case two here is p is not equal to a so p is not equal to a

and we’ll call this here case two so just say case two right there so now

we know that p is not equal to a well we um don’t know if p is equal to b or not

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but we don’t even need to think about that p is not equal to a so we have two

distinct points here so by axiom a a one

there’s a line that goes through them so there exists a line now here i named it

an l but you don’t even need to name it the name of the line is immaterial there

exists a line that is incident with p there exists a line and here i’ll just

name it it could be the same line but we don’t know there’s just a line m

incident with p because i have two distinct points axiom a1

and let’s justify this a little bit in terms of formatting and so

step 7 says that there are two two different points and then axiom a1 says

that there’s a line that passes through those two points in particular there’s a

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line that passes through uh point p here so um

in in either case we get a line that passes through p

no matter which case that we’re in um and so we’re and so our proof is done

here we have a line uh no matter which one holds p is equal

to a or p is not equal to a we have a line call it line l or call it line m but

either way there’s a line that passes through point p and so

um you know you could write a summary step right here

um all cases considered you know so there exists a line a line incident with p

some people uh really like this but some people don’t

they’re they’re just a line incident with p

and then you could just summarize you could say steps uh four and step um

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you know step six and step three so the three says there’s only two

cases so three six and then eight and this is the end of the proof because

this is um independent of or or taking into account

all the cases there exists a line incident with p

steps three says there’s only two cases step six is the end of case one step

eight is the end of step case two and so then the proof is finished right there

so there’s how you could write it up in a column format there’s one way to do

that um and so now let’s look at a paragraph proof

so the paragraph proof should have all the key ingredients

um and it should allow you to much quickly and easy easily write your

own pair column proof or line by line point of logic by point of logic

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uh or previous steps and so on all right so there’s going to be a proof here

suppose p suppose that p is a point that no line is incident with

that no line is incident with so by a3 by axiom a3 there exists three distinct

there exists three distinct distinct non-collinear points and we

didn’t really use the fact that they were non-clean here there exists three

distinct points say a b and c so i’m just going to say without loss of

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generality p is not equal to a so without loss of generality p is not

equal to a what does that mean well piece of point and we got these three

points that are different from each other so p is not one of them

and so i’ll just say it’s it’s a it’s not equal to a

now this is a summary because we know we took care of both cases it was equal to

a it wasn’t equal to a if it’s equal to a then p is not equal

to b so i’m just saying without loss of generality p is not equal to a and so by

a1 so by a1 we see that there exists a line that does pass through we see that

there exists a line incident with p incident with p

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contrary to hypothesis so you know we gotta because we said assume it’s

there’s no line right contrary to hypothesis and there we go so we assume that

there’s no line but we found that there is one so there is one there you know

every point is incident with one line right there and so there’s the proof

right there now um what i want to do now is talk about these things here

so let’s go to this one right here and this kind of gives us a summary

screen right here of what we’ve done so far

here’s our point line in incidence and here’s our three axioms here

and we’ve just proved uh theorem three right there

every point is on at least one line we went through it line by line and we

wrote a summary paragraph proof all right so uh

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in the next coupling of episodes we’re going to prove these two statements here

every line has at least one point not incident with it so every line

and then there’s some point not on it all right that makes sense we don’t want

all of our points to be on a line unless you’re doing some kind of

one-dimensional geometry but you know we’re trying to talk about points and

lines making up some kind of plane at least that’s our intuition but we’re

not using the word plane anywhere at all we’re only using these three words here

in logic so uh there’s four so five would be

sort of the dual of it replace line with point

every point has at least one line not on it

so every point there’s some line over here and so we’re going to prove those

in the next uh and upcoming uh episodes and you know right now so we proved

every point is on at least one line and then we’re going to prove every

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point is incident with at least two distinct lines so

we’ll do that upcoming so uh what i wanted to ask you about now are

um you know the difference between theorems and facts so when you’re

starting out something so young with such a small base of of assumptions here

you know what you call a theorem and what you call a fact or maybe a

proposition or a dilemma or something of that nature

you know really depends upon your frame of reference um but generally speaking

you know you may just call everything a theorem or maybe you want to

call something a fact so a fact i would say is something that is um you know

needs to be proven but also it’s not so very important it’s not very

enlightening something probably obvious perhaps

so um for example here’s a fact coming from these three uh axioms here

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um that you know we could write down as a theorem if we want but you know

do we have do we need to write a proof for it so for example here’s a fact

there is there exists at least one point right so it’s uh it’s a fact right a3

has there there exists three distinct points so knowing that a3 is true

you know at least there’s at least one point right so this is just a fact there

exists at least one point and the proof would just be you know by a3

in fact you could say another fact there exists

two distinct points right so again that would follow immediately by axiom a3 so

those are small little facts that fall immediately from the axioms um

and so you know i’m not going to write those out as theorems

but what i would like for you to do is to think of all the facts that you

can think of that uh whose proof are is immediate

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either from these three axioms or for or from these theorems here

and put it in the comment below and um yeah i’ll see if it actually needs to be

a theorem or if it’s just a fact and uh let’s do that and let’s see what you

guys come up with all right and so that’s it for this

episode here check out this one right here and i’ll see you in that episode