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hi everyone welcome back i’m dave um we’re gonna go over the incident

axioms and some basic theorems that we can prove using these axioms

so uh this episode is part of the series instance geometry tutorials with

step-by-step proofs the link is below in the description let’s do some math

so let’s go over the instant axioms first we’re going to let point line and

incidence be undefined terms so we’re thinking about incidence as being a

relation here for example a point lies on a line

or we say point is incident with a line and we have three axioms axiom one is if

you’re given two distinct points p and q then there exists a line that passes

through them in its unique line and the second axiom says

if you take any line you can find two distinct points on you can find two

points on it and the third axiom says there exists three distinct points with

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the property no line is incident with all three of them

and then we have our three basic definitions uh points are collinear if there is

exist a line that passes through all of them

lines are concurrent if they all have a point in common and in parallel

lines are called parallel if they have no points in common

so you can rewrite axiom 3 to say there exists three non-collinear points

which is a nice succinct way of looking at axiom three

in the previous episodes we have talked about uh so some these theorems here

we’ve proven theorem one and theorem two three four five and six

so we proved these six theorems so far so we’re going to prove some more

theorems in today’s episode um but unlike these other episodes we did

everything column format and then we wrote a paragraph proof

today we’re just going to write them in paragraph proof i’m going to write them

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in such a way that i hope that you’ll be able to write your own column proof uh

for them and and i heartily recommend that you do that

so before i prove each of the new theorems not d6 we’ve already done these six

i recommend pausing the video and trying to write a column proof yourself

and then coming back and looking at what i did uh now before we go on though

i want to go over some notation and a couple of other things

in case you haven’t seen the previous episodes

so for points i’m going to use capital letters so for example a b c

there’s three points there and when i write a b and c i don’t necessarily mean

that they’re different points so if i say if i mean that they’re

different i’ll say three distinct points and just because i say there’s three

points here that doesn’t mean that they’re all distinct from each

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other so just caveat there’s i’m very precise when i mean that they’re

different from each other um so if we um i’ll also use sometimes a p and

a q and an r for points also for lines i like to use lowercase

letters l m and n will be common notation for a line

um and then we have this right here so this is something that is really not

standard anywhere uh it’s just geometry is just

done by so many people it’s just hard to make a standard so

by axiom a1 if we’re given any two points say p and q

there’s only one line going through them or if i have say points a and b there’s

only one line going through them if they’re two distinct points

so um how the question is how to uh you what’s the notation for the actual

line right because we’re not using diagrams for notation right so i want to say

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e any one of these three right here the notation for them

so this is the notation i’ve been using so far it represents um

the line passing through two distinct points a and b

so that that’s that’s what this right here is notation for

um the line the unique line redundant right the unique line

uh incident um incident with a with incident the unique line incident with

distinct points a and b so as you can see that this notation right here

uh represents all of these words and so this notation is very handy the unique

line incident with two distinct points a and b

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and so i’ll use this right here also with a comma sometimes

and then sometimes without a comma the comma is really unnecessary

but some people like to have it but anyways all three of these are

notation and they just are representations of these words the

unique line incident with two distinct points a and b

um l is a unique line passing through two distinct points a and b

so uh that’s the other thing i wanted to talk about was

um you know there’s other ways of saying incidents

and so i’ve been i’ve been doing this but i just want to make it uh you know

pointed out so we can say uh p is incident or i’ll say um a is incident a is

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incident with line l all right and so that’s the word the

official word is incident and i say also a lies on line l and sometimes

i won’t even say point point a lies on line l a lies on l um

or we can say l passes through a where line l passes through point a okay so i

mean the same thing for all of them they all mean the same things just a

different way of saying just intuitive ways of saying the word incident incident

so we want to be rigorous as much as possible but we also want to have

intuition as much as possible we want everything don’t we all right so

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let’s go on now uh the fourth thing for today is going to be theorem seven like

i showed you a minute ago we’ve already proven theorem one through six we did

column format first in the previous episodes and we did the proofs and so

today right now i’m just going to work on the proofs here and i’m hoping that

you’ll pause the video right here and try to write out your own column proof

and i’ve shown you how to do that kind of six times so you know if you haven’t

seen those episodes go back and try them and try to write your column proof so

your column proof should have statement justification and you should number each

and every line and so what i’m going to do here though is just actually write up

a proof here all right so hopefully you paused already so here’s the proof here

so i’m going to use axiom a1 and it’s going to happen that these two lines are

going to be equal to each other now the reason why these two lines are going to

be equal to each other is because c and a are distinct so why are cna distinct

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right so c is uh distinct from a right so it says right there c is

distinct from a um c and a are on this line for sure and

they’re distinct from each other axiom a1 says there’s only one line that goes

through cna because cna are just two distinct points

but cna are also on this line right a is on this line and our hypothesis is that

c is on this line so c and a also go through this line but

axiom one says there’s only one line that goes through this line so axiom one

tells us immediately that these two are equal because i know that c and a are

distinct and they’re incident with both lines so those lines have to be equal to

each other by a1 so there’s my first step

so by a1 i’m going to use the same logic but instead with c and a i’m going to

use c and b so by a1 so line through the line through b and c

is also equal to the line through a and b so why is that because c and b are

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distinct right and it’s telling us if c is distinct from b right so so uh b and

c are obviously on this line but they’re also on this line b is on this line

right here but it says that c is on this line right here also right there so b

and c are on both of these lines so by a1 these lines have to be equal so when

i say by a1 right here i’m justifying this equal sign right here uh by by

saying this right here um and so that tells us that this line is equal to a b

and this line is equal to a b and so we have exactly what we want

all three lines have to be equal to each other

and so you know the intuition behind that is if c is on this line right here

so we have a line given by this two distinct points here a and b and it sees

somewhere on that line and it’s not a and it’s not b

so c could be here here here you know wherever it’s at but the line going

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through c and a is equal to the line going through c and b

which is equal to the line going through a and b they’re all the same line as

long as c is somewhere on that line um yeah so uh let’s go to

um nick’s theorem here now theorem eight all right so again pause the video and

try it out and so here’s going to be our proof uh if

these two lines are given and b and c are distinct points

then the lines through a and b is equal to the line uh through b and c

so here’s our proof so as you might imagine again we’re going to use a1

to get some lines are equal to each other so the line through a and b is unique

and how do we know it’s equal to the line through b and c

well first off we know there’s a line through b and c by a one there’s a line

going through it and it’s unique so b and c are on this line

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b is on this line right here um and we’re we’re assuming that these two

lines are equal which means where c is on this line so c is on this line

so in fact b and c are on both of these lines so by a1 they have to be equal

right so that’s how i say that here by a1 it follows these two lines are equal

there’s a unique line that goes through b and c

beings here on both of these lines here all right so then oops

so that’s the proof of uh theorem eight there it’s very very short there

it follows by by a1 right there all right let’s try another one um theorem 9

so if l is any line let’s just draw this here if l is any line

and let’s see here we’ve got a line so then there exist lines m and m

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and they’re distinct m m m n are not the same

but they both uh have a point in common with l so you know like here’s an m and

here’s an n so i have to be able to find those lines

right there how can i find two lines that are incident with this arbitrary

line l all right so here’s the proof so i recommend pausing it and trying to

write it column proof again all right so first thing i’m going to do

is right so l is any line so i’m going to say l is a line

uh now we have a previous theorem remember we already proved theorems one

through six in the previous episode so the m4 says

if you take any line you can find some point not on that line

so let’s just keep a diagram going over here here’s a line l and by theorem 4

so there’s some point p right here not on that line l it’s important that it’s

not on that line l all right and so there exist points a and b incident with

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l let me move my p out of the way that sounded funny um there exist points a and

b on line l okay there’s a and b uh why is there exist points a and b

uh incident with line l actually i forgot to

put that in here this is theorem nine i like to be as complete as possible so

while you’re thinking why that happens i’m going to fix this here okay so

all right sorry about that typo there uh or not typo but just uh oversight there

so by a2 axiom 82 remember a2 says no matter uh any line you have there’s two

points on it so i have this line here l and axiom a2 says there’s at least two

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points on every line all right and so now you might imagine

how we’re going to come up with the lines m and n so um

now we we’re going to know that a is not

p why is a not p well p is not on line l a is on line l right so i i say right

here a is not equal to p but the way that you would write write that up in a

in a column format is you would say a equals p um r a a hypothesis

and then you would say oh well then that puts p on line l because because a is

but p is not on line l so you get a contradiction and then you would say a

is not equal to p and you would get an r r a a conclusion from that right so i’m

skipping those uh small little tiny steps there that i think that you can

try to fill in for yourself so a is not p and then b is not p also right if if p

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was equal to b then p would be on line l but you know line l was made uh so i

mean p was made so that it was not online now okay so so these are not

equal to each other so what that means is by axiom a1 right there’s a line going

through a and p and there’s a line going through b and p

right now what we know is that these are distinct points a is not p and

b is not b and a and b are incident with line l and

p is not instant with line out so i tried to justify that a little bit there

so i didn’t skip any steps there all right so by a1 and i’ll move my

diagram down here again so we got line l we got p not on the line and we got our

a and b on the line here as as we know by axiom a1 there exists

lines through these two points here so there’s a line going through here which

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is line m and this is point p right here and there exists a line going through p

and b which i’ll just put right there and this is line n right there

so by a1 we can we can draw a line through a and p

which i’m i’m just going to call it l that’s what i mean by that notation

right there just call it just call it m and there exists a line through b and p

and i’m just going to name it you know it exists by the axiom so i just give it

a name and both these lines have a point in common with l

so what is the point in common with l right so a is on line l and a is on this

point uh and a is on this line and b is on line l and b is on line m in

right so both of these lines have a point in common with l

now the question is are they distinct right and so then here i say clearly l m

and n are distinct lines so um you know why do i say that why can’t they be the

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same well this is again where you would try some r a a hypothesis in your column

proof so you would say something like um you know they are equal all right so if

they’re equal right well uh p is on line m um and if they’re equal then p is um

um online n uh and there’s nothing wrong with that so let’s see here

if they’re all if both of these lines are equal to each other then

they’re all equal to each other because uh a is online m

and if they’re equal to each other that would put a on line in

and if they’re equal to each other that would put b on line m

and so l m and n would all be the same line and then p would be on that line

right so we cannot have these right here that would tell us that a and b are on

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all three lines making all three lines equal to each other by

a one here okay so again i i suppose you can

fill in those small little steps there clearly they’re distinct lines if you

try to set them equal to each other you will

run into a contradiction let’s just say that all right and so let’s look at uh

theorem 10 now um again try to pause and see if you can get the proof there

so if a is any point so there exist points uh sorry a is any point

uh then there exist points b and c uh such that there’s no line going

through all three of them there’s no line going through all three of them

they’re non-clean year so let’s see how to prove that

so i’m going to start off with a is a point and now remember by theorem 5

there exists a line non-incident with a so there’s a line here which i’m just

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going to call it l and we know by axiom a2 that every line has two distinct

points on it and i’m going to name them b and c

there exists distinct points b and c and i’m going to use these two points to

satisfy the theorem um and their incident with line l right

so every line has two points on it two distinct points

now what happens if we uh think that there’s one line going through all three

of them what’s so wrong with that so i’m

going to assume for a contradiction that

they are co-linear and i’m going to call

that line m so m is the line that passes through all three of them

well if it passes through all three of them it passes through b and c

but axiom a1 says there’s only one line that passes through b and c

so if l so if m is a line that passes through all three

it certainly passes through b and c and so it has to be equal to l

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but a was a point not incident with l right so there’s no way that there can

be a line going through all three of them so this contradiction shows right

the contradiction is that a is incident with l and a is not incident with l

so that’s a clear contradiction so we cannot have a line going through all

three of them so in fact they cannot be collinear which means they are in fact

non-collinear all right so there’s a theorem 10 and we got one more theorem here

it’s a very similar theorem so this time we have two points so let’s

say we have point a and b and let’s call it a and let’s call it b

um then there exists a point c such that a b and c are non-linear

so i have a point c here there we go and they’re non-collinear

so how we come up with point c well pause the video now and give it a try

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all right here we go let a and b be two distinct points

and so there exists a line um a through a b and

i didn’t say why there exists a line through a and b and i usually like to be

thorough so i’m gonna go back here just very quickly

and so that there there should be a bi theorem one here or axiom one right

okay got it all right here we go so by axiom a1 uh a1

is that a one or an exclamation mark here we go so by uh by a by axiom one by

a one there exists a line through a and b all right good we’re going to call

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that line l and and by theorem four they’re just a point

not incident with this l so i’m going to call that point here c

so by theorem four we know that there’s some point not on the line there all

right so i’m going to assume for contradiction now that i got these three

points i’m going to assume that there’s a line that goes through all three of

them and i’m going to call that line l now we’re hoping that’s not true but we

got to see that it’s not true what if it is true all right so if i get a line

that goes through all three of them so by a1

um the lines are equal and why why well because a and b are two distinct points

on line l and m also goes through a and b so

there’s only one line a1 says there’s only one line that goes through a and b

so they have to be equal and so that means that c is on line l

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but c was not on line l so this contradiction right here shows that this m

cannot exist in other words they cannot be collinear

so they must be non-collinear here we go

all right and there we go we have a kind of a summary right there of what we’ve

done so far um we did theorems one through six in

previous episodes and then in today’s episode we did seven eight nine ten and

eleven and i thought i would just put them all on one screen for us for us to

see there all right so uh what we got going on next is

parallelism yeah check out this episode right here to uh do some more math and

i’ll see you in that episode