Math Is Fun Forum
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#26 2014-03-08 00:42:23
Re: Cyclic Quadrilateral
Bob's solution looks intriguing. I'll see it.
BTW is it okay to say cos(x) = -cos(180-x)
What reasoning?
'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'
I'm not crazy, my mother had me tested.
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#27 2014-03-08 00:49:29
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Cyclic Quadrilateral
Yes, you can say that.
1) Graph it for 1 period.
2) The identity.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#28 2014-03-08 00:50:09
Re: Cyclic Quadrilateral
So by cosine rule on ABC
How are you doing this one?
'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'
I'm not crazy, my mother had me tested.
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#29 2014-03-08 01:06:56
- Bob
- Administrator
- Registered: 2010-06-20
- Posts: 10,643
Re: Cyclic Quadrilateral
hi
So if A = pi
cosine rule
It looks like this approach will generalise for any x, y and z.
Later I'll see if In can solve analytically.
Bob
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you! …………….Bob 
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#30 2014-03-08 01:17:38
Re: Cyclic Quadrilateral
Thank You. Analytically, what is that like?
'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'
I'm not crazy, my mother had me tested.
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#31 2014-03-08 02:02:19
- Bob
- Administrator
- Registered: 2010-06-20
- Posts: 10,643
Re: Cyclic Quadrilateral
I mean without using a mathematical program, rather doing 'by hand'. When I was young the only computers ran on valves and filled a large warehouse. I got used to simplifying and solving on a piece of paper. I'm not a luddite about using computers, but, sometimes, I find it satisfying to complete a question that way. I haven't tried it yet, but I'm optimistic that I can solve that cubic. Watch this space.
Bob
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you! …………….Bob
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#32 2014-03-08 02:10:40
Re: Cyclic Quadrilateral
If it is a cubic and has a perfect integer solution, I believe it can be done by factorisation or something. Isn't there a formula for cubic equations?
Which space?
'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'
I'm not crazy, my mother had me tested.
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#33 2014-03-08 02:27:34
- Bob
- Administrator
- Registered: 2010-06-20
- Posts: 10,643
Re: Cyclic Quadrilateral
see picture below.
B
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you! …………….Bob
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#34 2014-03-08 02:48:20
Re: Cyclic Quadrilateral
So, you like the default theme.
Sorry, the arrow only describes a point precisely. You did not tell me how much space around it you want to include. I mean, what are the boundaries of the 'space' you are describing?
'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'
I'm not crazy, my mother had me tested.
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#35 2014-03-08 03:09:46
- zetafunc.
- Guest
Re: Cyclic Quadrilateral
He means he is going to respond to your post with one of his own
#36 2014-03-08 03:12:32
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Cyclic Quadrilateral
You are in the wrong thread. That confused me.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#37 2014-03-08 08:35:34
- Bob
- Administrator
- Registered: 2010-06-20
- Posts: 10,643
Re: Cyclic Quadrilateral
Continued from post 23.
To save having to keep re-typing it I shall use the following substitution:
So the equation becomes:
So how many solutions will this have? I'm assuming we want d > 0 and in order to stand any chance of doing this without computing power, I'll also assume that d is an integer (+ve).
Consider the cubic
As the d cubed term is > 0 this cubic 'starts' in the third quadrant and goes to the first quadrant. Next differentiate:
This will have two turning points, at d = ± √ 31027
The first diagram below shows the general shape for the cubic.
Now consider the cubic:
This is an 'odd' function ie. it has rotational symmetry order 2 about the origin, and the same turning points. Even without a calculator (as I know that 27 x 27 = 289) I can say that the positive root of this cubic is > 270. I've labelled this cubic 'a' on the second diagram.
Now 'our' cubic ( labelled 'b' )is just a vertical translation of the second cubic. As you can see in the second part of the diagram, this means that d > 270.
Furthermore we can see that there will be only one solution in positive integers, so, once I've found a solution, it will be the only solution. (My sketch shows no negative solutions. It may be that I have not translated the right amount and that other solutions exist but they need not concern us.)
Suppose d is even. Then d^2 - 93081 is odd. So it cannot contain any power of 2, so d must be 2^8 times maybe more factors. There are 13 possible cases.
Now suppose d is odd. That means that d can only have odd factors. There are 9 possible cases.
So try each case until a solution emerges, then stop as that's the only one.
I find that d = 5 x 5 x 13 = 325
check: 325 x (325^2 - 93081) = 4076800
This method can be used for any values of x, y and z, although it may not always lead to a simple cubic. There is a formula for solving cubics in another thread so that should 'button this up'.
Bob
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you! …………….Bob
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#38 2014-03-08 20:25:58
Re: Cyclic Quadrilateral
He means he is going to respond to your post with one of his own
joking, I was.
Hi Bob,
Thanks
I'll look at it a little later.
'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'
I'm not crazy, my mother had me tested.
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