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in this episode we’ll write our first proof in incidence geometry

i’ll guide you every step of the way as if this was your first proof in geometry

you’ll see how to write a column proof and a paragraph proof and i’ll explain

the advantages with each approach while this theorem may not seem that

exciting it’s the underlying process that can lead to new ideas about the

world around us let’s do some math [Music]

hi everyone welcome back we’re going to first begin by recapping the incidents

axioms uh we went over these in axioms um at the the first episode the

beginning of the series and so here we go we have the first of

the axioms here a1 um for every point p and for every point q not equal to p

there exists a unique line l that passes through or is incident with p and q

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so recall that these three words right here point line and incidence

are undefined terms and we’re going to explain the behavior

of these three words right here by using the axioms and there’s three of them

so the second axiom a2 for every line l there exists at least two distinct

points incident with l and the third axiom recall

that there exists three distinct points with a property that no line is incident

with all three of them now if this is a little bit um

confusing um i go through this uh in detail and the uh series incidence geometry

um and that first episode so i just wanted to mention that the link is below

in the description to the full series but you definitely want to check out

that first uh video where i go over the incident axioms in greater detail

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so um we i also talked about uh what collinear are three three or more points

are choline here if there exists a line instant with all three of them

and three or more lines are called concurrent if there exists a

point incident with all of them and then the last definition that we

went over was lines l and m are parallel if they are distinct you know not equal

to each other and no point is incident with both of them

so we went through all of this in that previous episode and i just wanted to

refresh our memory in this episode though we’re going to do our first proof

we didn’t do a proof in the previous episode

and so here is our theorem our first theorem

can we prove it right so theorem one if two distinct lines are not parallel

right i’ll just sketch that out right if they’re not parallel then they have a

unique point in common and so our goal is to prove this so we have line l we

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have line m and we’re going to try to prove that

there’s one point on both of them and only one point on both of them

and like i said we’re going to do this uh in a column proof

and in a paragraph proof after we do the column proof here

so the column proof will be line by line just justifying each and every step

and what we can and cannot do i walk through all the logic rules

and how to work with quantifiers and i also talked about the other series

that you might want to look at besides this one to help you get prepared for

this but let’s go on now and look at our first proof here we go we’re going to

prove this theorem right here again this doesn’t seem very exciting in

itself it’s the underlying process that we’re concentrating on this

in this geometry right here so we have a statement and we have a justification i

usually write my statements on the left and then i’ll justify them over here on

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the right and so our first statement is for

you know if we have two distinct lines right so this is going to be our

hypothesis we have line l and line m and i’m going to assume that they’re not

equal to each other now the convention that i’m using is lowercase l m

i’m using those for lines and i’m using uppercase p and q and r and s and t

for points that’s just my convention that i’m using there if you’re ever

not following that or lost or disoriented you know

line l and line m are not equal you know we could always write that in words i

just chose to write it concisely like this

but anyways that’s our hypothesis right we have an if we have a then

so we have two distinct lines right here we go

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and we’re also assuming that they’re not parallel so l and m are not parallel so

also recall in that previous episode where i talked about this notation right

here for parallel that’s also hypothesis right

all right and so now we’re going to talk about

not parallel what does that mean right so we have a definition of parallel

and we have the negation right because we’re saying not parallel so that means

the negation of parallel right so there exists a point p that lies on l and m

so not parallel parallel means they have no point in common and if that’s not

true that means they well they have a point in common so p lies on l and m

and that’s true by the negation of parallel

all right and so number four step four here

there existed point q incident with l m and it’s not p

and that’s going to be my r a a hypothesis here see because i want this

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point to be unique i already have p lies on both of them where you know that

but now i’m going to want it to be unique and one way to show something is

unique is by using our a raa hypothesis and

assuming you have another one suppose there’s another point and by another

point i mean not equal and they’re both and this point is

incident with both lines so that’s going to be my ra hypothesis

now whenever you do an ra hypothesis that a goal is to get a contradiction

so i’m going to get a contradiction and since this is theorem one the only

things we can really use are our logic rules and our three axioms and so now i’m

going to use an axiom right here so axiom a1 recall says that

if you have um two distinct points if you have two points which we do we have

p and q we have p we have a q there’s only going to be one line going

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through them only one line we know l and m go through p

we know l and m go through q so a one says that they’re actually

equal to each other so now we have a contradiction between

steps one and step five and so this is going to give us a

contradiction right here now whenever i’m going to say i have a

contradiction right here it’s i’m going to be very clear with you

i’m going to i’m going to have this statement and its negation right

here or this statement and the negation right so you ha it has to be very clear

and obvious contradiction here um all right and so now

we have the negation of the ra hypothesis which is called the raa

conclusion right so here we said you know there’s a point q that’s not equal

right so actually q is equal to p and so p is unique and

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that’s the conclusion that we can draw from this right here and so

now what i’d like to show you is you know now that we’ve gone through this

line by line by line and justified each and every step

i’d like to show you like how to write this in a paragraph proof

the the goal is that this one right here is focusing on rigor and you know

depending upon you know your requirements every time you apply a rule

of negation or a logic rule you may want to um you know

write out even more steps but the point is is that this is focusing on rigor

when we try to prove this again we’re going to write this up in

paragraph proof and the idea behind a paragraph proof

is to convince the reader it’s not necessarily to go line by line in every

single excruciating detail it’s to convince the

readers to have all the important steps in it to convince the reader

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and so you have to take into account who your audience is

this approach right here is not taking into account your your audience well

your audience is not your focus what your focus is is rigor and

using your previous theorems your previous axioms and your logic rules

that is the focus whereas when we have a paragraph proof now

so now we’re going to have the same theorem theorem one if two

distinct lines are not parallel then they have a unique point in common so

now i’m going to write this up in paragraph form and the focus should be

on the reader and you know who is the reader if your reader is well versed in

incidence geometry then this is an obvious statement and they should be

able to prove it on their own if your audience is someone who’s never

seen this proof before then you will want to be as clear as possible

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and show as many details as possible and really give a nice structure and give a

nice you know reading so let’s go with our proof here

i’m going to suppose that l and m are distinct lines so i’ve actually named

the lines l m that way i can talk about them

notice that we don’t need to say that in here we can get away with that without

mentioning l m so you know it’s nice if you have a theorem that’s written up in

a nice catchy way where you can remember

it easy right but your proof should have all the gory details right

so since l m are not parallel right so um this is

you know conveying to somebody that we’re look that we’re looking at what

this right here means we’re applying that to what we already have

so since they are not parallel there exist a point incident with both of them

say p and now i’m going to let q be a point sorry

so i already got these two lines they already have one point on them

and i’m going to say hey what if there’s another point on them right we’re in

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incidence geometry we don’t know anything about the word straight

what if there’s another point on here right so let q be a point incident with

both of them and assume they’re not equal right so i got

two points on this line this line is called l and this line here is called m

but axiom h a one says lines and points don’t behave this way

if you have two distinct points there’s only going to be one line going through

them these lines have to be equal to each other l and m have to be equal to each

other by axiom a1 so by axiom a1 it follows that these are equal to each other

which is contrary to hypothesis right suppose they’re distinct lines

and so actually this assumption right here that p is not equal to q

leads to a contradiction and so p is equal to q

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and so there in fact is only one point in common

and so that does it for this theorem here and

in the uh next episode we’re going to look at another proof

and uh well that’s it for this episode so i hope you have a great day and i’ll

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