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## #1 2007-08-20 21:16:21

JaneFairfax
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### Geometry problem

Here’s a geometry problem I’ve just made up by myself.

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.

I’m sure it works. Still, if you find any problem with my problem, do let me know.

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

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

## #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.

igloo myrtilles fourmis

## #3 2007-08-22 21:42:40

JaneFairfax
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### Re: Geometry problem

That’s right. I should have provided a diagram anyway, so here it is.

|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?

## #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?

## #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?

## #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.

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

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

## #10 2010-11-23 00:06:07

bob bundy
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### Re: Geometry problem

Hi Jane Fairfax,

Bob

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.

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

## #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

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