Every Point Is Incident With at Least One Line

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

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

00:00
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

00:01
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

00:02
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

00:03
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

00:04
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

00:05
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

00:06
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

00:07
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

00:08
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

00:09
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

00:10
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

00:11
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

00:12
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

00:13
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

00:14
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

00:15
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

00:16
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

00:17
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

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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