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#1 2007-08-20 21:16:21

JaneFairfax
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Geometry problem

Heres a geometry problem Ive just made up by myself. big_smile

Let O be a fixed point. Let Q be a variable point such that the length of OQ is less than or equal to 2a, where a is a fixed positive real number. Now let P and R be points satisfying the following conditions:

(i) OQPRO is a rectangle.
(ii) If PQ is shifted parallel to itself to P′Q′ such that P′Q′ and PR mutually bisect each other, then PP′RQ′P is a rhombus with sides of length a.

Prove that the locus of P is a circle of radius 2a.

Im sure it works. Still, if you find any problem with my problem, do let me know. tongue

Last edited by JaneFairfax (2007-08-20 21:20:35)


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.
 

#2 2007-08-22 11:05:32

John E. Franklin
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Re: Geometry problem

Here are six red points.
Is that what you mean?
I only used up 279 bytes
on that small pic!!
The lower-center red dot
is Q prime, unmarked as such.


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View Image: rhombus.png      


igloo myrtilles fourmis
 

#3 2007-08-22 21:42:40

JaneFairfax
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Re: Geometry problem

Thats right. I should have provided a diagram anyway, so here it is.

http://i188.photobucket.com/albums/z198/fairjane/GeometryProblem.png

|P′Q′| = |PQ|
|PP′| = |P′R| = |RQ′| = |Q′P| = a
|OQ| ≤ 2a

Note that P can be below OQ as well, and that Q can also be to the left of O. Indeed, Q can be anywhere within a distance of 2a of O but for the purpose of analysis, we can assume WLOG that OQ is horizontal.

Last edited by JaneFairfax (2007-08-22 21:52:07)


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.
 

#4 2007-08-24 21:55:49

JaneFairfax
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Re: Geometry problem


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.
 

#5 2007-08-28 06:59:48

John E. Franklin
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Re: Geometry problem

What's a locus?  Is it a geometry term?


igloo myrtilles fourmis
 

#6 2007-09-18 22:41:10

landof+
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Re: Geometry problem

Definition of a locus

That may be of some use.


I shall be on leave until I say so...
 

#7 2010-04-06 11:18:24

JaneFairfax
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Re: Geometry problem

Bumping this because after two and a half years nobody appears to have a solution yet.


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.
 

#8 2010-11-22 03:51:33

123ronnie321
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Re: Geometry problem

I am using your figure to solve this.

let the midpoint of PR be M
Let OQ = t < 2a
Let PQ = k.

http://i188.photobucket.com/albums/z198/fairjane/GeometryProblem.png

PQ = P`Q` = k = 2P'M
PR = OQ = t = 2PM
PM^2 + P`M^2 = P`P^2 = a^2        ...  Pythagoras theorem

therefore, k^2 + t^2 = (2a)^2. which is a circle if we make the following assumptions-
Let OQ act like x axis  and and OR as Y axis.
Co ordinates of pt P are k,t.

Last edited by 123ronnie321 (2010-11-23 23:49:12)

 

#9 2010-11-22 09:47:59

bob bundy
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Re: Geometry problem

Hi Jane,

I haven't been ignoring your problem since 2007.  I've only been a member for 6 months or so.

So thanks for bringing back into attention.

See my two diagrams below.  I set up what I thought was the right diagram and took the first screen shot.  Then I moved Q left a bit and all that happened was moved the same amount (second half of shot).

Would you mind repeating the construction rules.

Mystified Bob dunno


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#10 2010-11-23 00:06:07

bob bundy
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Re: Geometry problem

Hi Jane Fairfax,

Ok, so I hadn't read the problem properly.  Now I have.



Bob


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Last edited by bob bundy (2010-11-23 00:07:53)


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
 

#11 2010-11-23 22:28:09

JaneFairfax
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Re: Geometry problem

Hi, Ronnie and Bob. smile


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.
 

#12 2010-11-24 01:43:02

bob bundy
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Re: Geometry problem

Hi Jane

I thought I'd already proved it in post 10.

Bob  smile


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
 

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