Hex Myst 11

We apply our understanding of cross ratios to prove a conjectured formula relating 5 points on a Pappus line. This brings out a curious duality between two points. We will also look at using GeoGebra to figure out algebraic relationships between quantities.

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0:02

Hello everyone i'm Norman Wildberger and thanks for joining me on this adventure

0:06

we're looking at Pappus theorem and some interesting relations having to do with

0:10

cross ratios and uh and two interesting points that we found so we're going to use what we've

0:17

learned about cross ratios and apply it in the situation but we'll also have an opportunity to

0:24

discuss a very interesting trick to sort of discern mathematical relationships using

0:29

geogebra that'll be quite interesting so there's a very hands-on approach we're we're tackling

0:35

geometry but we're doing it in an algebraic way and it's a very concrete uh you know explicit um

0:44

uh approach to the to the subject which i want to encourage you to think in this direction

0:49

i want you to think less in terms of abstract you know let this be this and now let be this this be

0:55

this you know that kind of sort of abstract stuff that's based on intuition intuition and ultimately

1:02

dubious logical foundations which is rampant in modern pure mathematics rather what we want to do

1:07

is we want to really be solid in looking at very specific things and and we can bring out you know

1:15

what's going on with algebra so lots of lots of interesting things happening so let me remind you

1:22

the general situation we're talking about Pappus theorem we've got one axis horizontal another axis

1:29

vertical we've got three points a1 a2 a3 on the horizontal axis there they are and b1 b2 b3 on the

1:36

vertical axis and then there are these interesting quantities that depend on the coordinates of these

1:41

six points okay and the the yellow Pappus points that we've discovered that are really

1:47

quite interesting say t and u have these rather simple expressions in in terms of these rather

1:53

complicated things and then the line that's joining t and u we'll call it sort of the pink

1:59

Pappus line has a relatively uh simple expression also like this or or like this and in particular

2:08

that allows us very quickly to deduce the uh the uh the x coordinate for such a um a point so uh

2:18

the x intercept is then r so if we set y equal to 0 we get x equals 2y over z the y intercept is s

2:27

0 comma 2x over w and we're interested in this r point here and y over z if we go back up there

2:35

it's just an expression purely dependent only on the a i points right okay so now we want to

2:45

deal with this cross ratio conjecture let me just remind you you have four collinear points

2:50

a b c d cross ratio of those four points it's a number it's a ratio of ratios so we're taking the

2:57

vector ac the vector bc looking at the ratio of those vectors

3:02

we're also looking at vector a d over b d and that ratio and we're taking the ratio of these ratios

3:09

now in terms of just a one dimensional situation where our points have coordinates a b

3:13

c d this is same as c minus a over c minus b divided by d minus a over d minus b and we can

3:23

sort of refl flip the uh the second fraction to write it as a quotient a minus c times b minus d

3:28

over a minus d b minus c okay so let's uh first of all we'll clear our definitions to start with

3:37

clear definitions and we will define this as our cross ratio r all right now

3:44

this uh interesting this interesting uh

3:50

conjecture that we're supposing that o is the meat of the two Pappus lines l1 and l2 okay with l1

3:58

containing a1 a2 a3 and and then r is the meat of the pink Pappus line with l1 and we're interested

4:06

in the relationship between this point o and this point r there seems to be some remarkable duality

4:12

between these two points and we can capture that numerically by this relationship between three

4:17

cross ratios cross ratio c1 c2 c3 cross ratio of a2 o a3 r so we're taking two out of these three a

4:29

i points there's an o and an r answer cyclically c 2 is r of a 3 o a 1 r and c 3 is the cross ratio

4:40

of a 1 o and a2 r and we want to establish this pretty interesting relation that two times the

4:48

sum c1 plus c2 plus c3 is the product c1 c2 c3 so we have a ultimately a cubic relation

4:58

amongst all these things okay so what are we going to do well this r point here this uh

5:05

this this point r has x coordinate uh we've just seen it it's it's two times this quantity here so

5:12

let's call that little r okay so little r is going to be uh this quantity here and we can

5:22

okay we can actually define that all right okay and

5:30

then we can calculate these cross ratios okay so the cross ratio of a2 a2 is just

5:42

the number a you know the coordinate of the capital a2 0 which is the coordinate of o

5:48

a3 which is the coordinate of a3 and little r which is the coordinate of r okay so we just

5:54

calculate this so let me actually sort of i've already done it but a little let's uh

5:58

if i just evaluate this okay so that's that's that's the expression as a cross ratio

6:05

and the complexity all comes about because of this r thing and then if i say simplify this

6:14

then i get this thing here which is the same as

6:17

as this thing here okay so i'm just calculating this cross ratio

6:25

and i've also calculated the the second one c2 there it is there similar kind of expression

6:32

and c3 similar kind of expression all right so now well now let's define these quantities so that's

6:40

c1 that's c2 that's c3 so i'm going to define these things let's define them define define

6:49

define okay great and now i can you know attempt to see what happens um

6:59

let's look at two times c sub one plus c sub two plus c sub three

7:07

okay just the sum of these various things notice that the denominators are not exactly the same

7:12

so if i take the sum and say say simplify i'm going to get this expression here let me see

7:18

if i can factor it that actually simplifies it further yeah that makes it quite simple well

7:25

there's a bit of simplification so let's have a look at that there it is there that is the twice

7:33

the sum of these three cross products so there is this quad quartic term actually cubed on top

7:40

there's product of the a i's product of the differences of the ai's and then there are these

7:44

three other factors okay now let's have a look at c1 c2 c3 and let's just factor that all at once

8:01

looks very very similar doesn't it so let's just check

8:05

let's check equality how can we do that we just form an equation so this thing equals this thing

8:21

and

8:26

we check equality and it is true so we have proven our

8:34

cubic relation by going to coordinates writing everything out getting our computer to simplify

8:39

left and right hand sides and seeing that they're actually the same very satisfying

8:48

okay so let's have a look at geogebra so this is the same kind of situation

8:53

we have three points on the horizontal axis here a1 a2 a3 and i've actually moved this other

9:02

axis so it's attached to the variable point o and this points b1 b2 b3 on that and our yellow points

9:11

here they are there there's one there and there's one there there's that pink line which is the join

9:16

of the yellow points and the meat with the x axis here is this point r okay and we're interested in

9:25

these five points r and o and a1 a2 a3 which sort of determine the relationship between o and r so

9:34

i want to explore the relationship between o and r we've already sort of looked at it um you know

9:43

sort of qualitatively like this but i want to be a little bit more precise about trying to ferret out

9:49

what the relationship between these two things is so what i'm going to do here is i'm going to

9:54

first write down the coordinates the actual the x coordinate of both points so let's say x of o okay

10:04

that's uh quantity 5.52 is that actually correct yeah here's point o up here and this x coordinate

10:10

is indeed 5.52 i might call it something else though let's rename it let's call it

10:19

little o that's really consistent what we've been doing before

10:23

right now same thing with r so let's take the x coordinate of the point r

10:31

and that's minus 0.71 let's just check where is r here r is here it is minus

10:39

0.71 yes it's the same so similarly we will rename this so let's call it little r

10:50

okay we're interested in the relationship between o and r now that calculation that we've just done

10:56

with the the cross ratios you know does give us something but i want to just sort of find it uh

11:02

geometrically here so this is now an important technique that i found useful on many occasions

11:08

i'm going to share with you all right so i want to understand the relationship between these two

11:12

things okay so i move you know this the o and the r is also changing in some fashion so what i'm

11:18

going to do is i'm going to create a new point okay whose coordinates are o and r these two x

11:27

coordinates so for the two points i'm interested in so i'm just going to introduce that point

11:32

a where is it it looks like it's right here let's make it a little bit more noticeable

11:42

okay color let's make it uh let's say blue

11:45

and star we have to make it bigger there we go all right

11:52

so there's the point a so what does this point a do so this point a is capturing at the same time

11:59

the coordinate of o as well as the coordinate of r okay so let's just watch what happens to this a

12:09

it's moving along some reasonable curve can we get at that curve a little bit more clearly

12:14

yes we can if i click on this thing here and i'm going to put a trace on okay that

12:21

means well you'll see what it means as i move all around this point a is moving

12:27

but jojoba is recording sort of where its trajectory is so so let's go in this direction

12:36

okay that looks like uh something of a hyperbola just keep moving oh look there's a

12:41

there's another branch my goodness this looks like a really does look like a hyperbola doesn't it

12:54

so hyperbola is quadratic it's a quadratic relation it looks like there's a quadratic

12:59

relation between o and r okay um so this is a very very useful technique uh to to remember and uh so

13:10

now let's go back to our to our algebra and see if we can see this quadratic relation between o and r

13:22

okay so um i want to sort of encode what we've been talking about so we found an interesting

13:28

kind of phenomena so let's introduce a definition that sort of captures um to a certain extent what

13:35

we're what we're finding so the essence of it is that we have these five points on the x axis so

13:40

three a points a one a two a three and these other two points o and r okay and we found

13:45

a certain relationship between them in terms of cross ratios so in general suppose we have points

13:51

o which is a general point uh with coordinate x coordinate oh zero and r's are a zero and a

13:59

one is a one zero and a two is a2 0 and a3 is a30 okay so these are just the coordinates o

14:05

r a1 a2 a3 of these five points let's say that these five points form a Pappus quintuple okay

14:14

if the three cross ratios exactly the ones we've been looking at c1 this cross ratio here okay

14:21

a20a3r which in in general this will just be this thing here okay and c2 same kind of thing a3o a1r

14:30

it's going to be that thing there c3 red cross ratio a1 0 a2r that could be that thing there

14:37

if these three cross ratios satisfy this relation that we found two times the sum

14:42

equals the product okay so let's investigate this let's see what this actually means

14:51

so here we have these three cross ratios and i can

15:00

define them

15:04

so see that they're there there's c1 c2 c3 fine and oh i've got r already there

15:13

oh i want to i don't want to have r so i'm going to first of all uh clear definitions

15:20

okay and let me do that again so now c1 r is just a general points not a specific point okay so now

15:28

i have there are those three cross ratios c1 c2 c3 and i'm interested in this in this condition

15:35

you know this is the condition that we found that this difference 2 times the sum minus the product

15:42

is going to be 0. so let's actually compute this okay because these are all cross ratios so

15:48

we're going to compute it and of course we want to simplify this so let's simplify it

15:54

so what do we see well there's some factors um kind of nice o minus r squared on top and then

16:01

there's these six different factors basically differences between o's and the a points and

16:08

between r and the a points and then there's this additional factor okay so let's have a look at it

16:16

so let me isolate that again it's sort of running off the page i can put it all on

16:21

the page by asking my computer to expand this out okay so now i've expanded it out

16:30

and now i'm interested in thinking about this as a relationship between o and r so we're thinking of

16:36

a1 a2 a3 is sort of constants that determines the situation it's really the relationship

16:41

between all and r that we're interested so i'm going to rewrite this polynomial collect

16:49

o and r okay so let's have a look what we get so we get some something here times

16:57

o r plus something else times o plus something else times r plus just

17:03

uh something else involving no r's always so we get a we get a quadratic relation between

17:15

o and r so this is verifying that that hyperbolic trajectory that we found using geogebra

17:26

but we have now more specific information because we have exactly what this equation is and

17:34

if i introduce these quantities say y is this one here z is this one here u is this one here

17:48

then this thing here the first thing here is is basically uh after we divided by 2 is is u

17:56

this quantity and this quantity here the coefficients of o and r are both the same and

18:03

that's basically a minus a half of z and this last thing after we divide by two it's just y

18:11

so let me remind you that this this u z and and y um these ones here i'm i'm

18:16

not just uh introducing something really new they're already contained here at least the y

18:22

is and the z is we've already seen them okay however the u is is a little bit different

18:31

we we've introduced this this new thing u okay and here we get this symmetric relation um

18:41

that's giving us the quadratic relation between o and r so this kind of quadratic thing um can

18:49

be described also in terms of the matrix this can be described in terms of a matrix a 2 by 2 matrix

19:00

let me show you that 2 by 2 matrix so we put a u up here we put a minus half zen here it's

19:13

going to be a symmetric matrix and would put a y here so for example if i uh let me just take a

19:24

make a column vector here i make a column vector uh say with r and one and i make a row vector

19:36

with

19:39

o and one okay and i evaluate this thing i get um y yes r o u that's nothing there minus a half z of

19:56

r and minus a half so we get the same thing that's that's that's a true fact okay so this

20:02

relationship equaling zero can be expressed in terms of this

20:06

this two by two matrix so this two by two matrix uh kind of captures the the um the quadratic

20:13

the conic that we're talking about whenever you have such a two by two matrix you should also

20:18

check the determinant the determinant this two by two matrix is u y minus one quarter z squared

20:29

i'm going to now define these again i had just undefined them so let's define them again

20:37

so they're back to what they were in terms of the ai's

20:42

let's just evaluate this thing say simplify

20:50

have a look at that is that beautiful

20:54

right so this um this relation that we're getting even though it's looks

21:00

quite complicated in terms of the entries of this two by two matrix the determinant is very simple

21:07

it's just that and while we're at it we should also compute the trace just to

21:14

because that's the other kind of invariant of the 2x2 matrix so

21:17

let's go to matrices let's calculate trace let's simplify that if it does

21:25

it doesn't simplify okay fair enough but the determinant does and the determinant

21:30

somehow um you know has something to do with with what kind of conic what we're getting

21:36

so i think we've kind of closed the circle here we have established in fact yes that there is

21:41

a quadratic relationship between o and r and it's symmetric remember we observed that that

21:46

the o and the r playing a symmetrical row we can see that very clearly here because

21:50

it's a symmetric expression in o and r and okay it's non-trivial it involves these

21:59

these functions of the ai points so it's a little bit uh a little bit non-trivial but still there's

22:06

some some pleasant algebra because the determinant of this thing is is very very nice just uh three

22:12

quarters times the the squares of the difference of the various ai points very pleasant.