Geometry Challenge: Two Lines and One Ruler

btw, can you answer the last

question? if you dont forget your physics.

which has more mass?

For the circle is very simple Take center to P as a diameter and make a circle where it cut circle A are the 2 tangent points. For the ellipse is more complex I have to think it over when I shall have the time .

Don't you think it is simple ?

take the ruler and fix the edge along point P and rotate it till it is tangent to the ellipse ?

haha, you seem to be beyond math.

Hoever, you win an encouragement prize.

Perhaps, Euclid wasn't able to translate the definition of a straight line into Chinese?

Very good. come on.

I shall vote your answer a full good.

The second is somehow complex. Its belong to college level. beyond middle school.

He he, kris is kris.

you solve the problem at last. I vote .

However, I dont really believe its your work. Im afraid you may copy from some web site. although I dont know where it was.

Im afraid you cannt prove out. Can you ?

I have also two point guesses, which I shall explain laterly.

After all, your english man is stronger than some america teachers.

The only thing to do, since you gave no circle centre, is to draw rays from the point. Doing so doesn't indicate anything that seems useful at first, but as soon as you joint the intercepts (which is the only thing you can do with a ruler), the solution becomes clearer.

I figured it out myself, and drew it on a small utility. Here it is, with the labels still showing (they pop up automatically during construction).

As a solution, I think it looks to be correct, and can probably be shown as true with one of the circle theorems (too many to list. nine-point/pascal etc etc).

Yes, I just checked in a book, and it appears to be Pascal - the construction effectively forms a hexagon inscribed on the circle. "The three points of intersection of pairs of opposite sides are collinear". It wasn't in my mind at the time, but the way you presented the question inevitably lead to that construction. With just a ruler, you can do no more than draw rays from 'P', and also through any intercepts that follow.

You can't prove it was copied, because it wasn't. My result, which follows Pascal as described, is the only path one can follow in looking for a solution. Maybe Pascal arrived at his theorem by simply drawing lines from a point through a circle. I've no idea, but I certainly didn't realize what I was doing at the time. Believe what you want.

This may be not a copy, but I believe you might not uderstand this procedure when you DIY with some small software like geometry pic etc.

you only guess it was. becuase it has nothing to do with 9 points circle. it may has someting to do with Pascal law, however it will complex to prove it with this law.

Now you know it can be done by onely a ruler. this is a universal method to draw a tangent, not only circle but ellipse.and sspread to other curve.

I said you copy, becuae 1. I see you draw an extra ray and extra point. In fact that extra piont is a tangent from K.

extent KL line you will get another tangent of P.

I shall put out the beautiful picture next.

I hardly thought it s a so hard job here. Originaly I thought there must be many engineers can draw this graphics. espeically machine engineers.

"you only guess it was"

I'm beginning to think that the lights are on, but nobody is home !

Maybe the 9-point circle is a bit like your famous remainder, or even like the question and your subsequent comments - misleading.

How do you think Pascal/Brianchon/Whoever got to derive their theorems ? It could be ;

1) Cheating

2) Alien mind probes

3) They read your post and went back in time

4) They liked to doodle/think.

The construction I posted was what logically follows from your question and original drawing.

You are making wild guesses, and assumptions about my understanding. Using the software is just an alternative to scanning an image of a picture drawn by hand.

"but I believe you might not uderstand this procedure when you DIY with some small software like geometry pic etc"

In what way does that statement not apply to your original post ?

I could have elaborated on the solution, adding inscribed and circumscribed hexagons, but it would be difficult to view. What you got, was a solution I derived to the problem as posted. It was only by looking into your (poorly worded) question and comments that I found the solution. Briachon etc was not in my mind when I found the solution. By looking at your question I arrived at Briachon, not the other way around. The question did not prompt me to consider such theorems.

You are too cynical, thinking that others reach for text-book solutions. A great deal of my pure mathematics knowledge is from simply using blank paper and pen. There's a big difference between reading about some theory, and learning by investigating geometry. I call it understating/comprehension. Perhaps in China you 'learn' by repeating something 100 times, I don't know. If you were more enlightened, you might have the grace to acknowledge other peoples work.

You cannot claim that anyone was first to prove 'Pythagoras' theorem.

When you post your beautiful picture, please tell me who's software you use and who's work you are copying. I'm interested to know if you solved this problem yourself (as I did), were told the solution, or spent hours looking for relevant theory in a text book.

you seem to look upset. haha, I just have a guess. neednt take it seriously.

You are smart to do these. I dont think I could do it if I deduce it.

I know that wonderful picture. Chinese found it 3000 years ago. you can also find it in the book in Han dynastic about 2000 years ago. But today most of scholar called it as pythagoras theorem

I hvnt got a ruler now. I draw it by autoCAD. other sortware can also do it. even a MSpaint can do.

jujst draw line or ray. what you do is making it looking beautiful.

I have no ability to invent such solution. it was only an exercise in the old times. just like draw Pentagon in geometry class. Can you draw it?

If you would know pole, polar, four points figures you may know what book I shall mention. Im afraid few people here learned such subject.

The Pentagon* is just the Golden Section in another guise, so there are many ways to draw it. Starting with a square is 'best', but there are lots of ways to draw √5.

I haven't any idea what book you like, but would be interested to know.

*Better not say more if this was to do with Rockets etc. The Americans know just as well how to convert a Square. We have a nice old blockhouse near ours;

wow, you are smart. I really forget that there is a Pentagon Building in USA.

Golden Mean can be use for anywhere in the life and work. especially in fine art.

the building in the pic looks like a old facfory.

Not so smart, it's just that we all have our own areas of interest/passions and collect bits of 'trivia'.

The building is usually called the 'Admiralty Blockhouse'. It's 1940/ cold war-era, located next to Trafalgar Square in London. "It was built 1940-41 as an operations centre for the Admiralty. The foundations are nine metres deep, and a six metre thick concrete roof protects the principal rooms. It remains a gaunt reminder of the Second World War. In an attempt to disguise the original function of the building, it is now covered with Russian vine and the roof is grassed over, and is cut twice a year.". Plenty more on the internet.

Nice !

I believe that a proof for the circle is readily available on the web. Naturally, the single line will find both tangents.
Extension to ellipses, parabola and hyperbolae can be made by projecting between planes cutting a cone.
A question: does anyone know of a reasonably simple 2D proof (preferably available on the web) that extends to hyperbolae?

I believe that a proof for the circle is readily available on the web

Not you as bloody well !

That was not intended to imply that you cribbed it - the sole object was to write about the extension to other conic sections, which is (to my mind) rather elegant.

You will find a trail here.
BTW, isn't it quite surprising that this one was as recent as Pascal...

Gosh, did I forget to add a '' ?

As I pointed out in #40 to cnpower, a defence against such a suggestion is hardly needed;

"The only thing to do, since you gave no circle centre, is to draw rays from the point. Doing so doesn't indicate anything that seems useful at first, but as soon as you joint the intercepts (which is the only thing you can do with a ruler), the solution becomes clearer."

As it happens, I have a jolly good book which covers such stuff. Although it appeared mad at first, cnpowers question is a good one. As we were discussing over on the Challenge Question, it's good fodder for helping students to arrive at some basic theory for themselves.

Okay, I'll bite...they both have the same mass, namely one kg.

I don't have the answer to the geometry question, but nick name cannot be right because the distance from P to the center of the circle is not the same as the distance from P to the tangent point.

Once you make sense of the concept of tangent line, you will know nick name is quit right at the first question.

Draw a circle through P and center of the circle, that means all points on the new circle and lines link each of them to these two points P and center of existing circle will have a right angle. so do the two cross points of the two circles. connect p and the new two points, they are tangent lines of the existing cirlcle.

Thats all. but you have to use a compasses in this way.

The second is really somehow complex. However I give him a half of good answer now :)

Hi cnpower, Could you please present your explanation above in a sketch, as I agree with ba/ael that the method proposed by nick name is not correct.

You are absolutely right, This is not a tangent

Simple

let us say the Circle origin = O and the point of intersection on circle (cn's tangent point) = C

then

PC=OP

ie

angle POC=PCO

But to be tangent PCO=0o deg

hence POC=90 deg

Hence CPO = 0 deg

ie the circle diameter = 0 or the circle is a point.

The sketch

Actual method is

Principle is very simple, We know that the system will form a rectangle (or a right angled triangle)

The hypoteneous is the dist OP, one arm is the radius of the main circle, the other one is the tangent.

sb,

First you need to locate the circles origin. That's easy enough, just swing arcs on the preimeter etc, blah blah, etc.

Next, you draw the line OP

Next, you have to find the point C, such that OCP is 90^o (wherever the p it is). You can do it here, (sorry, but much better than gremnlin drawing) and focus on elipse here. Maybe you can get P taken from here too. Today is p-taking day (more than normal, anyway) so I'm not logging in. Not worry about GA's on this stuff, it goggles easy enough.

Kris

Is the Guest E_man? Why not log in ?

haha, this may be more interesting.

There are another two inner cross tangents. totally 4 tangent lines.

I still vote yours as good answer.

you can use any other ways to implement the ellipse tangent. In fact, only one ruler can be done.

My mistake...nick name is RIGHT after all. But he needs a compass doesn't he? And Euclid did allow a compass.

Sorry, I'm not quite sure on the rules (rulers?).

Are we allowed to use a pair of compasses.

Is the ruler just a "straight edge", or, can we use it "measure" things. (I'm not absolutely certain but I think a ruler with perfect measurement capability would be equivalent to straight edge and compasses.)

Yes, its a straight edge ruler, no scale, no number on it.

Only use one ruler. dont allow to use compasses.

[In fact, ruler means thsi straight edge things. cannt measure in practice geometry drawing]

==

ps.

I dont think this is a challenge. I still like original title: if you havent forgotten math, try to draw thes tangent lines.

Correct :

somewhat (=\= somehow)

dont allow to use compasses

Are you sure?

Chinese circles/elipses come with premarked centres/focul points. A swivelling ruler is not a compass. Don't know how, it just is. Maybe it's all drawn on graph paper and they use a calculator. Think I'll back the origami solution now.

I may guess who you are. haha.

neednt to lurk.

press one end of the ruler then you can draw a cirlcle use antoher end. as well as a rope. Man on the earth will know this way. hehe.

cannt help to wait for your wonderful origami work.

haha, I have no idea how to say " a pair of compasses" ?

it s used for drawing a circle, not for pointing direction in the field.

how to say this compasses?

I know you home at math, have a try, I shall give you a GA, if its right.

My twisted mind went straight to getting the Queen of England or someone like her to draw your lines. Then you have one ruler drawing both lines.

Now that's thinking out of the box!

Can the paper be folded, so when unfolded one line would appear as two lines. Ha you could also get the Queen to do that! LOL

Dear Cnpower,

It is hard to understand Your question because perhaps You chose wrong English word.... I cannot see why You ask if it could be done with just ONE ruler (meaning straight edge, I hope), since with one ruler we can draw as many lines as necessary.

Then You first say there is two circles, and if it is so then ellipse is just circle in another plane. As someone has suggested, we can position straight edge to touch point P and circle, and wherever line touch circle from outside this is tangent to this circle, and same rule is used for making tangent to ellipse (or other circle, but then if line drawn is not on same plane as circle, does it qualify to be called tangent? I think it does....) and we dont need to lift ruler from paper also. If we draw from circle to point P and from point P to ellipse, then we dont need to lift pencil and then this is technically one line :-))

But You refer to Euclid geometry which perhaps is just distraction that draw attention from simplest solution.

Next, I remember some similar problem where paper should be folded so that edge (line) of circle and line of ellipse touch in some point and then folded another way so that edge goes from line of circle to point P and from line of ellipse to point P. Holding that two edges together we can use side of long pencil tip to draw line on both edges at same time, thus drawing two lines at once in same move of pencil... Of course, this is not Euclid geometry method, is that what You mean?

Answer to last question is that mass is same. This is quite similar to trick question teacher asked in 3rd. year of Primary school: >>What weight more, 1 Kg. of feathers or 1 Kg of lead?<<.........

Now, it seems that You think nobody answered Your question correctly? Is that so?

er, you can draw as many lines as you can by a ruler, but from point P can only drawe two tangent line to a circle, as well as ellipse.

A:

You can do as sb did, from P at one end of the ruler, shift it until it contact the circle, then draw line. it might be a tangent. Of cause, it might not be a tangent line, but a secant line. Becaue when enlarge your pic, you may find out it. You may argue, then I draw, enlarge, draw ... well, at last, I can do the trick.

I admit you could do it, just as do like ancient calculated the area of a circle., but math is a serious job.

B:

So look back sb in his #15, fr the second pic, you will understand. what is tangent from a outside point P and what is math.

C:

ellipse is a special circle when parameter a equal to b.( parameter equation)

If I said they are all conic curve. you may ask me, Hi, where is cone?

D:

If I have time I shall tell you what is drawing tool in Euclid geometry. If you think you would have a good basic math, I can tell you what is Euclid system, what is axiom. Of cause its beyond of a middle school range.

E:

Last weenend I met a Ca retired middle school math teacher who had teached in USA fro many years. I asked him come here for this solution. I though its no hard for him to solve.

I havnt seen his thread, though.

F:

the third question is,

a piece of iron and a pack of contton, [ placed on the table in students class room], weight 1kg severally(each) . Which is the most mass?

ah, so many people read this thread, but leave no message.

I try to survey, do you think its possible or impossible?

please contribute your idea with yes or no

Y or N.

Nickname has already found the answer for the circle.

For the ellipse, draw a circle with diameter lying directly over the major diameter of the ellipse. Use nickname's method for finding the tangent point of the circle. Finally draw lines perpendicular to the major diameter of the ellipse. Where they cross the ellipse will be the two points of tangency.

Hi,

I cannt imagine whats your idea. Can you illustrate?

In some speicial condition, yours may be found. but generally speaking Im afraid it will not be found.

the correct answer sees kris at his #37 thread. Although he leaked out a line (it should be two tangents, he has drawed only one) [thats why I said he may not prove out and that was his copy]

I've added a few more lines, and dragged the point 'P' and it's rays around a bit. I deleted the tangent. You can find new ones, now that Ive drawn the hexagon, so it's no big loss

Nice utility to play geometry on, isn't it. Trouble is, I'm not going to tell you what utility it is !!!! ha ha haaa. You shouldn't have suggested I cheated. I'm sure you can find plenty of similar utilities - this one is a simple download that's fun for playing Euclid. It can add in all the angles if you want to check stuff.

cnpower : To keep your specific problem in accordance with Briachon's theorem, you need to ignore K, and complete the hexagon with points on the arcs EH and FG (at J, and the opposite circle intercept). That would be hard to see in the above picture.

K will work fine to illustrate the theorem, but you will be looking at concurrent points for the heaxagon DHLGFB (ie, fewer lines are added to illustrate the general case).