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        riddles >> easy >> sun-tan-gent
        
(Message started by: Noke Lieu on May 30th, 2010, 11:47pm)
Title: sun-tan-gent
Post by Noke Lieu on May 30th, 2010, 11:47pm
Quite around here at the moment (this forum). I blame the northern hemisphere comign into summer, and folk leaving the safety of inside...

Given two different circles, separated by a distance greater than R1+R2; construct the common cross tangent.
Title: Re: sun-tan-gent
Post by towr on May 31st, 2010, 2:06am
See attachment.
Although I've assumed the centers of the circles were given. But they're constructed easily enough.
Title: Re: sun-tan-gent
Post by TenaliRaman on May 31st, 2010, 6:45am

on 05/31/10 at 02:06:07, towr wrote:
See attachment.
Although I've assumed the centers of the circles were given. But they're constructed easily enough.

[hide]Is it a well known property that the cross tangents are concurrent with Line CD ??[/hide]

-- AI
Title: Re: sun-tan-gent
Post by towr on May 31st, 2010, 7:58am

on 05/31/10 at 06:45:16, TenaliRaman wrote:
[hide]Is it a well known property that the cross tangents are concurrent with Line CD ??[/hide]

[hide]Say CD and AB intersect at X; triangles ACX and BDX are similar triangles. Say Y is the intersection between tangent 1 and circle 2 (and E is the intersection with AB), then triangle AGE is similar to BYE.

So we have
|AE|/|AG| = |BE|/|BY|
|AX|/|AC| = |BX|/|BD|
from similarity of triangles, but also,
|AG| = |AC|
|BY| = |BD|
since they are the radius of their respective circle. So therefore |AE|/|BE| = |AX|/|BX|, so X=E.

It works as long as angle BAC = angle ABD (by the same reasoning); right angles were just the easiest choice for me.
[/hide]
Title: Re: sun-tan-gent
Post by Noke Lieu on May 31st, 2010, 7:07pm
:D

It's funny how coming up with the riddles takes more time than the solving! I'll keep scratching around to see what I can see...
Title: Re: sun-tan-gent
Post by towr on Jun 1st, 2010, 12:37am
It took a long enough time to make that picture.  ::)
Title: Re: sun-tan-gent
Post by Noke Lieu on Jun 1st, 2010, 5:04am
I must commend you on your new toy- they're beautiful pictures.
Title: Re: sun-tan-gent
Post by towr on Jun 1st, 2010, 6:04am
It's a combination of two toys, actually (and neither very new); Cinderella (http://www.cinderella.de/) (1.4) for making constructions, and Inkscape to finish it.
Title: Re: sun-tan-gent
Post by TenaliRaman on Jun 2nd, 2010, 7:16am
Thanks for the proof.

My question was more along the lines of, whether you were already aware of this property. Because, it looks like the key to the construction you gave.

@Noke,
I don't think anybody could solve problems as fast as towr does :-). Atleast, I can assure you that I was still racking my brain when towr posted his solution.


on 05/31/10 at 07:58:18, towr wrote:
[hide]Say CD and AB intersect at X; triangles ACX and BDX are similar triangles. Say Y is the intersection between tangent 1 and circle 2 (and E is the intersection with AB), then triangle AGE is similar to BYE.

So we have
|AE|/|AG| = |BE|/|BY|
|AX|/|AC| = |BX|/|BD|
from similarity of triangles, but also,
|AG| = |AC|
|BY| = |BD|
since they are the radius of their respective circle. So therefore |AE|/|BE| = |AX|/|BX|, so X=E.

It works as long as angle BAC = angle ABD (by the same reasoning); right angles were just the easiest choice for me.
[/hide]
Title: Re: sun-tan-gent
Post by towr on Jun 2nd, 2010, 7:48am

on 06/02/10 at 07:16:13, TenaliRaman wrote:
My question was more along the lines of, whether you were already aware of this property. Because, it looks like the key to the construction you gave.
I wasn't really aware of it before, it just came to me; I figured the ratio AE to EB should be the same in both cases. And I hadn't really thought it trough until you asked before.
The thing that gave me more trouble was figuring out how to construct the two tangents using that point.
Title: Re: sun-tan-gent
Post by Noke Lieu on Jun 2nd, 2010, 7:53pm

on 06/02/10 at 07:16:13, TenaliRaman wrote:
@Noke,
I don't think anybody could solve problems as fast as towr does :-). Atleast, I can assure you that I was still racking my brain when towr posted his solution.


;D Too true!
That said, it's why I'm glad to come here. The number of really gifted folk that have drifted by these dark pages has left me inspired.
It used to be easier to get questions when the boards were busier, but even empires wax and wane. I don't doubt that one day we'll return to hive of activity, but I suspect that we may need to have more people posting wrongish answers so newbies aren't so intimidated.
At least, that's my excuse!
Title: Re: sun-tan-gent
Post by TenaliRaman on Jun 3rd, 2010, 11:32am

on 06/02/10 at 07:48:43, towr wrote:
The thing that gave me more trouble was figuring out how to construct the two tangents using that point.

LOL, I would have never thought that would have been the case :-)

http://www.mathopenref.com/consttangents.html

-- AI


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