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#1 2007-08-19 23:16:21

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Geometry problem

Here’s a geometry problem I’ve 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.

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

Last edited by JaneFairfax (2007-08-19 23:20:35)


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#2 2007-08-21 13:05:32

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,582

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.

View Image: rhombus.png

igloo myrtilles fourmis

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#3 2007-08-21 23:42:40

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Geometry problem

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

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 (2007-08-21 23:52:07)


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A: Click here for answer.

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#4 2007-08-23 23:55:49

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Geometry problem


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

A: Click here for answer.

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#5 2007-08-27 08:59:48

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,582

Re: Geometry problem

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


igloo myrtilles fourmis

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#6 2007-09-18 00:41:10

landof+
Member
Registered: 2007-03-24
Posts: 131

Re: Geometry problem

Definition of a locus

That may be of some use.


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

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#7 2010-04-05 13:18:24

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

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.

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#8 2010-11-21 04:51:33

123ronnie321
Member
Registered: 2010-09-28
Posts: 128

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.

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 00:49:12)

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#9 2010-11-21 10:47:59

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,536

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

View Image: Jane Fairfax geom.JPG

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

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#10 2010-11-22 01:06:07

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,536

Re: Geometry problem

Hi Jane Fairfax,

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

Bob

View Image: Jane Fairfax geom.JPG

Last edited by bob bundy (2010-11-22 01:07:53)


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

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#11 2010-11-22 23:28:09

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Geometry problem

Hi, Ronnie and Bob. smile


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

A: Click here for answer.

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#12 2010-11-23 02:43:02

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,536

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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