Math Is Fun Forum

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Circles #3

The drawing will make it clear but I can not get it in there.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Circles #3

hi Agnishom,

Alternate segment theorem?

Haven't used this for years.

EDIT: earlier error corrected.

In any circle, centre A, EF a tangent at F, D some point on the circle:

Let EFD = x              =>      FCD = x (5th angle property of a circle)

http://www.mathisfunforum.com/viewtopic … 88#p220488

Now consider the triangles EFC and EDF

angle E is common,  and EFD = FCE = x

So EFC and EDF are similar

=> ED/EF = EF/EC  => ED.EC = EF^2

Similarly ED.EC  = EG^2

=> EF = EG

Thanks Phro. 

Bob

Last edited by Bob (2014-01-30 06:22:24)

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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

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Circles #3

I think the diagram here does what you wanted in post 14.  I need to think some more about whether it is ok or  a cheat.  You see as the equal property holds there's a danger with trying to construct a diagram that genuinely shows this as opposed to one that just looks right.

Anyway here's what I did.

(i) Join A to E, construct the midpoint and hence construct a circle with AE as diameter which makes EF a tangent to the original circle.

(ii) Do the same with the second circle to get G.

(iii) Find H the other point on the first circle which will make a tangent EH.

(iv) Bisect angle HEG.

(v) Find J where this bisector cuts BG.

(vi) Construct the circle with J as centre and with EG and EH as tangents.

Then EF = EH and EG = EH => EF = EG as required.

Bob

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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

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Circles #3

No it doesn't work.

I repeated the construction but with E not on CB.

The diagram is somewhat complicated so I've tried to colour code it.  The dotted circles are just there to fix the points F and G.

H is the second tangent point as before.  The red circle does have both the line EH and the line EG as tangents but not both of G and H are the actual tangent contact points.  I've chosen to make G one such point and you can see that H is not the other.  Big red arrow.

This just shows how careful you have to be proving things.  The result is true when E is on CB so that diagram looks ok.  But this diagram shows that the construction is in-valid.

Bob

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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

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Circles #3

That is not a big deal, we can change that line to: Draw a circle touching EH and EG and H and G only.

(It is still cheating, because we do not know if that can be done.)

What about bobbym's proof? He still did not tell me why those points must coincide.

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,895

Re: Circles #3

Hi Bob,

Ggrr.  If only I had gone surfing.  But the sea temperature is around 9 C.  Ugh!

The water temperature is very bearable down here!

But the air temp isn't! We've had quite a few 40+ days lately, so the aircond's been working flat out (and thank goodness it's working!!)

The beach pics are ones I took of Bondi (Sydney) last November.

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"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

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Circles #3

Nice scenery, looks like Florida.

What about bobbym's proof? He still did not tell me why those points must coincide.

I think that is the easiest part, they are constructed to coincide. See the other thread.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.