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In a trapezium ABCD, if AB||CD then AC² + BD² = BC² + AB² + 2ABCD

How?

*Last edited by Agnishom (2013-01-19 16:48:11)*

'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'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,469

hi Agnishom,

I haven't fully explored this yet; maybe later; but it has the look of several applications of the cosine rule. As no cos(X) terms exist they must be cancelling out. Remember that angle ABD = angle BDC because of the parallels.

http://www.mathsisfun.com/algebra/trig-cosine-law.html

So perhaps you could have a go yourself ?

LATER EDIT:

I've tried what I suggested and reached a formula connecting AC, BD, BC, AB, CD. But it isn't this one:

AC² + BD² = BC² + AB² + 2ABCD

So I set up the diagram on Sketchpad and got (for my trapezium)

LHS = 162.86 and RHS = 193.64

Are you sure you've got the right identity ?

Bob

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,998

Hi Agnishom;

How?

You might have a hard time proving that because I have been playing with trapeziums in geogebra and I can not find a single one where that is true.

Did you copy the problem correctly? Or what am I missing?

**In mathematics, you don't understand things. You just get used to them.**

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,469

hi bobbym,

Wow! Are we in telepathic communication or what ?

My edit was at your time less 20 seconds.

Bob

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,998

Hi Bob;

Looks that way!

I have been looking at it for a long time, I wanted to be sure. I thought maybe I was doing something wrong.

**In mathematics, you don't understand things. You just get used to them.**

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,469

hi bobbym

I thought maybe I was doing something wrong.

That's the difference between us then. I tend to blunder in and post anyway. I guess that's why my posts are full of corrections.

Four applications of the cosine rule does allow you to eliminate any angles and get an equation. It isn't that one; not even close.

That's as far as I've got.

Bob

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,998

Not so different. I blunder around plenty but Agnishom is usually careful.

**In mathematics, you don't understand things. You just get used to them.**

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Let me check....

'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'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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Sorry, it should have been **AC² + BD² = BC² + AD² + 2ABCD**

P.S.: I have a hint in my book. But I dont get it. Do you want it?

*Last edited by Agnishom (2013-01-20 01:08:14)*

'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'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 88,998

Hi Agnishom;

That one I believe works, yes post the hint.

**In mathematics, you don't understand things. You just get used to them.**

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Let CD < AB,

Drop perpendiculars DE and CF from D and C respectively.

In ΔABC, angle B is acute

Therefore, AC^2 = BC^2 + AB^2 - 2*AB*AE

In ΔABD, angle A is acute

Therefore, BD^2 = AD^2 + AB^2 - 2*AB*AF

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,469

Now my diagram isn't making sense. Is the lettering A to B to C to D ?

Bob

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**scientia****Member**- Registered: 2009-11-13
- Posts: 222

Hence

*Last edited by scientia (2013-01-20 08:26:55)*

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Got it .... Thanks

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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