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Geometric teaser. Construct tangens for two circles.


noobsaibot

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Common rules for this geometric teasers.

There is an initial figure (eg a circle), and there is something that should be at the end (such as a triangle in the circle). To solve the puzzles, you can only use a ruler and compass, respectively, you can build segments, circles and copy it. Nothing else

There are many teasers such this, but this was very difficult for me.


You should construct a tangent simultaneously to two circles so that in one circumferential line touching the top, and a second at the bottom. Please see pictures.

Original:

Screen_Shot_2014_10_28_at_23_17_23.jpg

Required result:

Screen_Shot_2014_10_28_at_23_17_09.jpg

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I'm not really sure what you're asking us to do. Could you elaborate on the puzzle?

 

It's a Euclidean construction puzzle. You get a straight edge that can draw lines, and a compass that can draw circles and be set to replicate any length already defined on page. You can do an incredible amount of construction with just these. You cannot however square a circle - this is the origin of that unsolvable, nor can you trisect an angle.

 

Common rules for this geometric teasers.
There is an initial figure (eg a circle), and there is something that should be at the end (such as a triangle in the circle). To solve the puzzles, you can only use a ruler and compass, respectively, you can build segments, circles and copy it. Nothing else
There are many teasers such this, but this was very difficult for me.
You should construct a tangent simultaneously to two circles so that in one circumferential line touching the top, and a second at the bottom. Please see pictures.
Original:
Screen_Shot_2014_10_28_at_23_17_23.jpg
Required result:
Screen_Shot_2014_10_28_at_23_17_09.jpg

 

 

 

1. Connect centres designated O and P

2. Mark from M (intersection of Circle A with OP) the length of the radius of B to create N

3. Construct new circle Circle C concentric with A with radius ON (ie R_C=R_A+R_B)

4. Using arbitrary Arcs create perpendicular bisector of OP - this crosses OP at point S

5. Set Compass to radius OS.

6. Mark circle A with intersection of Arcs with centre S and radius OS. Call these points T and T2

7. Join P (centre of Circle B) with Point T . This is necessarily tangent to Circle C

8. Construct line from O to T. This is radius of both Circle A and Circle C.

9. Where this line intersects Circle A mark point V

10. Construct line Parallel to TP which passes through V

 

This line is perpendicular to radius at point V ie is tangent to circle A and is parallel to a radius of Circle B and offset by exactly the radius of Circle B and is thus tangent to Circle B

 

Will post a diagram

post-32514-0-53927000-1414583386_thumb.jpg

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elfmotat,

Sorry for my bad description, imatfaal described very well above .

 

imatfaal, thanks for response.

I tried implement your solution but dont receive excepted result. I think i dont fully understand your naming.

 

My naming: point A - center of Circle A and point B - circle of point B.

I will show screenshots for each step(sorry for point names, it generates automatically and sometimes game overlay another points)

 

1. Connect centres designated O and P

 

image.jpg

2. Mark from M (intersection of Circle A with OP) the length of the radius of B to create N

 

image.jpg

3. Construct new circle Circle C concentric with A with radius ON

image.jpg

4. Using arbitrary Arcs create perpendicular bisector of OP - this crosses OP at point S

 

image.jpg

5. Set Compass to radius OS.

6. Mark circle A with intersection of Arcs with centre S and radius OS. Call these points T and T2

 

image.jpg

7. Join P (centre of Circle B) with Point T . This is necessarily tangent to Circle C

 

image.jpg

8. Construct line from O to T. This is radius of both Circle A and Circle C.

9. Where this line intersects Circle A mark point V

 

image.jpg

10. Construct line Parallel to TP which passes through V

 

image.jpg

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elfmotat,

Sorry for my bad description, imatfaal described very well above .

 

imatfaal, thanks for response.

I tried implement your solution but dont receive excepted result. I think i dont fully understand your naming.

 

6. Mark circle A with intersection of Arcs with centre S and radius OS. Call these points T and T2

 

This line of my description was typed incorrectly.

 

6. Mark circle C with intersection of Arcs with centre S and radius OS. Call these points T and T2

 

The diagram was correct

You should be able to see that T (top middle of page) is on circle C not Circle A. Sorry for typo

BTW what diagram making device are you using - it looks very neat.

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1. Construct point C on the biggest circle
2. Translate line AC to point B
3. Find the intersection of circle B and translated line (let it be point D)
4. Construct two rays AB and CD
5. Find the intersection of them->point E
6. Find the midpoint of AE or BE -> point F
7. Construct circle with center in F and radius FE
8. J is the intersection, ray JE is a solution

Do you have more such teasers?

Edited by anastasiia
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1. Construct point C on the biggest circle

2. Translate line AC to point B

3. Find the intersection of circle B and translated line (let it be point D)

4. Construct two rays AB and CD

5. Find the intersection of them->point E

6. Find the midpoint of AE or BE -> point F

7. Construct circle with center in F and radius FE

8. J is the intersection, ray JE is a solution

 

Do you have more such teasers?

 

Could you provide a diagram - I am getting lost with your construction

Aw you've made it too easy, I was under the assumption you did not have the circle centres available.

 

You said this was the first of a series.

 

Please make the specifications more precise next time.

 

 

draw any two chords

using arbitrary arcs greater than half length of first chord find perpendicular bisector

rinse repeat with second chord

intersection of two bisectors is centre of circle.

 

rinse repeat with second circle

 

continue as above

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imatfaal, i finally got it! Thanks for diagram, it was very helpful.

My result

Screen_Shot_2014_10_29_at_15_39_02.jpg


anastasia, thanks for you response.

 

I am try solve similar puzzle. It very similar to previous, but i cant understand how to solve it.

 

Source:

zad.jpg

Excepted result:

finish.jpg

 

BTW, i found this puzzle at http://geometricpuzzle.jff.name/ . I hope i dont break forum rules, when post external link, in another case i will remove link asap.

Edited by noobsaibot
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1. Form line AB from centre to centre

2. Let intersection of AB and circle B be called C

3. Set compass to radius of circle A

4. Mark this distance from C on Line AB - call this new point D

5. Create new circle C with centre at B and radius BD. (this circle has radius equal to radius_B less radius_A)

6. Bisect AB. Intersection of Bisector and AB is now called E

7. Set compass length to AE.

8. Draw arcs with centre E and intersecting with circle C. Call these intersections F and F2

9. Create line AF

10. Create line BF - and extend till it intersects outer circle B. Mark this point as G.

11 Create line parallel to AF which passes through point G

 

This line is both a tangent of circle b and a tangent of circle A.

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imatfaal, thanks, it works.

 

It likes so easy for you. Do you solve similar puzzles before? Can you share it for me?

 

 

When I was last doing these sort of euclidean construction the only method of sharing was copying out, putting in an envelope, and posting! So I am afraid I have nothing to share- but I am sure there is stuff out on the internet. I am not sure I would spend too much time - geometrical puzzle solving is fun but for a young man such as yourself I would venture that it is a tiny bit of a dead-end when it comes to actually learning; if you want an intellectual challenge then the internet has maths and science courses with fairly rigorous self-assessment in abundance.

Although I have just checked out your link and I will be downloading to my phone - far better puzzles than angry birds! and much more rewarding.

 

I didn't mean to put you off euclidean construction for fun diversion and learning - but I think in mathematical terms it is a little bit like latin; useful, a great foundation, the basis of many great past discoveries, but in the end a dead language.

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  • 2 weeks later...

I didn't mean to put you off euclidean construction for fun diversion and learning - but I think in mathematical terms it is a little bit like latin; useful, a great foundation, the basis of many great past discoveries, but in the end a dead language.

 

Why bother about the latin when you can read the english translations.

 

I have been reading "On the Shoulders of Giants", edited, with commentary by Stephen Hawking and it contains 'On the Revolutions of Heavenly Spheres' by Nicolaus Copernicus, 'Dialogues Concerning Two Sciences' by Galieo Galilei, 'Harmony of the World, book Five' by Johannes Kepler, 'Principia' by Isaac Newton and selections from Albert Einstein's papers on relativity.

 

They all invested substantial effort in developing their ideas geometrically before finalising the maths.

 

Incidentally, whenever I read Einstein's 1905 SR paper I get the distinct impression that he intended it to allow the true beauty of relativity to shine through a window onto the euclidean world. Unfortunately somebody put up the shutters.

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