Two Distinct Non-parallel Lines Have a Unique Point in Common

Video Series: Incidence Geometry (Tutorials with Step-by-Step Proofs)

(D4M) — Here is the video transcript for this video.

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

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

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

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

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

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

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

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

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

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

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

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

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
see you then if you enjoyed this video please like and subscribe to my channel
and click the bell icon to get new video updates

About The Author
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David A. Smith (Dave)

Mathematics Educator

David A. Smith is the CEO and founder of Dave4Math. His background is in mathematics (B.S. & M.S. in Mathematics), computer science, and undergraduate teaching (15+ years). With extensive experience in higher education and a passion for learning, his professional and academic careers revolve around advancing knowledge for himself and others. His work helps others learn about subjects that can help them in their personal and professional lives.

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