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#1 2014-05-10 21:38:37

maxpcs
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Registered: 2014-05-10
Posts: 1

prove that two circles have the same center

please help me to solve this problem!

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#2 2014-05-10 23:17:41

Bob
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Registered: 2010-06-20
Posts: 10,059

Re: prove that two circles have the same center

hi maxpcs

Welcome to the forum.

I confused about that diagram.  Here's what I understand.

DEF is an equilateral triangle, with H, K and V as the midpoints of the sides.  O is the centre of the inscribed circle for DEF.

Then I start to get confused.  There's a triangle ABC.  In your diagram it looks like it might be equilateral as well, but that may be wrong.  You've marked some lines as equal; AD, BE and CF.  The large circle is the circumcircle for ABC.

The radius of the smaller circle seems to be 'Y' and the height of DEF (KF) is Y + 2Y = 3Y.  But then you say DH = 1.5Y.  That won't work because that would make the sides of DEF equal to 3Y.  Hence my confusion.

Let's see if I can ask the right questions to sort this out.

Is ABC a special triangle, or may it be any?

Am I right that AD = BE = CF ?

Which length (by its endpoints) are you calling Y ?

Please post back.  smile

EDIT:  By experiment, it looks like ABC is forced to be equilateral.  Just trying to prove it.

Bob

Last edited by Bob (2014-05-10 23:55:51)


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#3 2014-05-11 05:43:27

phrontister
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From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,810

Re: prove that two circles have the same center

Hi Bob,

I've got the same problem with this as you.

The drawing seems to indicate that the green and red lines start at the centre of the small circle and end at their respective lettered points, but if that is so then all of those nominated lengths that are supposed to equal 1.5Y actually equal √(3Y²) instead.

I wonder if the OP, or maybe the puzzle setter, divided 3Y² by 2 instead of taking its square root to get the 1.5Y for those lengths.

By experiment, it looks like ABC is forced to be equilateral.

I drew this up in Geogebra and found the same, but only by experiment. Also that the circles are concentric, but I don't know how to prove it.

Last edited by phrontister (2014-05-11 06:21:26)


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#4 2014-05-11 07:37:37

phrontister
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Re: prove that two circles have the same center

Hi Bob,

Here is my Geogebra drawing.

I drew the easy bits first: The inner circle and its 3 radii @ 120° to each other, the circumscribed ΔDEF, and the 3 vertices of that triangle to its centre. 

Then I drew lines through D || HO, through F || VO and through E || KO, and formed the large triangle at the line intersections. Tidied it all up to look like the OP's drawing.

The 2 triangles and circles are all concentric - as the drawing shows - but I haven't found a way yet to prove that. Maybe you'd like to take over from here, as I have to go to bed (been watching the golf while I've been drawing, but must get some sleep now)...and you're much better at this than I am anyway. dizzy

Last edited by phrontister (2014-05-11 07:42:01)


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#5 2014-05-11 08:09:51

Bob
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Re: prove that two circles have the same center

hi Phro,

I'm using Sketchpad.  As you will see from my measurements, it's hard to get the exact values.

This is my construction method:

Draw DEF.
Choose a random point for A and make the line AD produced.
Choose a random B on that line.
Draw BE produced and AF produced to meet at C.
Measure AD, BE and CF.
Use trial and improvement, moving first A, then B, in order to make the three measurements equal.
Diagram below shows my best efforts.
Measured ABC and BCA.

The circumcentre for ABC and the inscribed centre for DEF are coincident.

Note:  In my diagram AD is not parallel to HO (nor the other two.)  The dotted line is parallel to AD.  It misses H.

There must be a way of proving ABC is equilateral
, after which I think it is straight forward to prove the centres property.

The underlined bit is proving elusive at the moment.  dizzy

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#6 2014-05-11 13:57:24

phrontister
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From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,810

Re: prove that two circles have the same center

Hi Bob,

Thanks for showing me your method, but I'll have to wait until this evening to look at it as I'm off to work now.

Btw, the drawing I posted is my second Geogebra attempt, and, unlike the experimenting I did in my first attempt (the one mentioned in my first post), I didn't need to do any manoeuvring of shapes this time.

Also:
1. Y (which I gave a value of 1) is the only length I used, with 2Y's length of value=2 happening automatically during construction);
2. I didn't use any angles other than the "3 radii @ 120° to each other" (which I could do as ΔDEF is equilateral...from OP's drawing), from which I drew tangents perpendicular to the radii to construct ΔDEF.

The symmetry of the construction prior to the large circle_large triangle stage triggered the idea that if I could base ΔABC's construction somehow on equilateral ΔDEF, the larger triangle might well also end up as an equilateral triangle...which it did! smile

And so I hit on the idea of making each side of ΔABC parallel to one of the height sides DV, EH and FK (those being the red+green lines, that result in straight lines).

Drawing those new sides through the 3 vertices (respectively) of ΔDEF give converging lines that meet at the vertices of new equilateral triangle ΔABC.

But the proof of all that escapes me...


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#7 2014-05-14 00:09:29

phrontister
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From: The Land of Tomorrow
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Re: prove that two circles have the same center

Hi Bob,

I had a look at your construction method in Sketchpad, and yes, I can see the 'trial and improvement' problem it has...which my first attempt also had.

bob bundy wrote:

There must be a way of proving ABC is equilateral.

I think I've got it. smile And this is how I can do it (I hope it's right):

After drawing the small equilateral ΔDEF and its inscribed circle, draw the following lines: through D perpendicular to DF; through F perpendicular to FE; through E perpendicular to ED. Those lines converge to intersect each other at points that are the vertices of large ΔABC.

Three new right-angled triangles - ADF, CFE and BED - have now been formed, one at each side of ΔDEF. They are congruent to each other, which means that ΔABC is equilateral as each of its sides consists of a hypotenuse and short side of the newly-formed triangles.   

The difference with this attempt compared to my other ones is that, after noticing in my post #4 drawing that ∠ADF looked like a right angle (and then measuring and proving it in my Geogebra file), I now force the right angle there, and also at the other two sides of ΔDEF.

Can you see a neat way of proving the two circles' concentricity from there? My way is messier than I like.

Last edited by phrontister (2014-05-14 05:52:50)


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#8 2014-05-14 06:52:16

Bob
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Re: prove that two circles have the same center

hi Phro,

You're practically finished.  smile

If you're right about the parallels and right angles then:

The circumcentre, centre of gravity and inscribed centres of any equilateral triangle are the same. (but not necessarily as each other yet).

Angle BDV = 60 so if DV produced cuts BC at say W, then BDW is equilateral, DA = WC and hence D is one third along BA => O is one third up the median => the centres coincide.  smile

But, ....., why is it ok to assume the parallel and 90 properties (one implies the other so I'm happy if either is justified).  In my version these did not hold.  sad

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#9 2014-05-14 19:18:42

Bob
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Posts: 10,059

Re: prove that two circles have the same center

Try this:

Draw an equilateral triangle ABC.  Use angle bisectors (or whatever) to find the centre.

Choose an arbitrary point D on AB.  With that as a point on the circumference draw a circle to cut BC at E and CA at F.

Draw DEF.  Check it is equilateral.

Mark the midpoint of DF as H.  Draw an extended line OH.  Check the size of angle ADF.

D can be moved along AB to achieve a variety of values for this angle and mostly OH is not parallel to BA.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#10 2014-05-15 01:13:27

phrontister
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From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,810

Re: prove that two circles have the same center

Ooo...interesting! That finds a second position (rotation) of the smaller inscribed equilateral triangle inside the larger equilateral triangle. I drew it up in Geogebra in much the same way as the post #7 drawing, and it all works.

With mine, the smaller triangle is rotated 30° anticlockwise from the larger triangle, while your smaller one is @ 60° anticlockwise. They'd be the only positions that would work, I'd say...at least with two equilateral triangles.

With yours I can see an easy proof of the circles' concentricity via medians and common centroids.


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#11 2014-05-15 10:27:02

Bob
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Registered: 2010-06-20
Posts: 10,059

Re: prove that two circles have the same center

Hi phrontister,

I have a complete proof with no angle assumptions.

Diagram below

I am assuming that ABC is any triangle; that AD = BE = CF; and that DEF is equilateral.

Let the acute angles shown be x, y and z.  With no loss of generality let's say x ≥ y ≥ z.  By which I mean if they're not equal choose the biggest to be x and so on.  I've shown BDE to be x, but the proof is unaffected by changing another to be x.

Sine rule in triangle ADF

and in triangle BED

But AD = BE and DF = DE so we have

As x ≥ z and both are acute, the above can only be true if x = z.

Similarly y is equal to both too.

So triangles ADF, BED and CFE are congruent and so AF = BD = CE and so ABC is equilateral.

Now let O be the circumcentre of ABC.

Angle DAO = 30 so by the cosine rule

Similarly

and

But AD = BE = CF and AO = BO = CO.  So  DO = EO = FO, => O is the circumcentre of DEF.

But in an equilateral triangle the circumcentre and the inscribed centre are the same so the inscribed centre of DEF is the circumcentre of ABC.

QED.  smile

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#12 2014-05-16 12:38:16

phrontister
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From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,810

Re: prove that two circles have the same center

Hi Bob,

Well, I couldn't have put that any better myself...and we both know why! dizzy

I started trying to nut out your proof last night, but it was too late for an honest go sleep and I'll have to try again when I'm more awake. Nice that you've been able to come up with a solve-all, though, that does away with angle assumptions! up

Meanwhile, contrary to this...

phrontister wrote:

They'd be the only positions that would work, I'd say...at least with two equilateral triangles.

...I've discovered that all rotational positions (I didn't try them all!!) of, say, the larger triangle in relation to a fixed-size smaller inscribed triangle, work, but the larger triangle's size needs to adjust to remain circumscribed. And verce visa...

In the example image, O is the centre of the two triangles and the two circles.

Last edited by phrontister (2014-05-16 12:39:51)


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#13 2014-05-22 22:50:39

phrontister
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From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,810

Re: prove that two circles have the same center

Hi Bob,

phrontister wrote:

They'd be the only positions that would work, I'd say...at least with two equilateral triangles.

phrontister wrote:

...all rotational positions (I didn't try them all!!) of, say, the larger triangle in relation to a fixed-size smaller inscribed triangle, work, but the larger triangle's size needs to adjust to remain circumscribed. And verce visa...

Here's a video showing both scenarios with equilateral triangles.


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#14 2014-05-23 00:10:41

Bob
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Registered: 2010-06-20
Posts: 10,059

Re: prove that two circles have the same center

Neat job.  Well done!  smile

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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