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#1 2014-02-09 19:23:31

thedarktiger
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sort of power of a point

Let \overline{PA} and \overline{PC} be tangents from P to a circle. Let B and D be points on the circle such that B, D, and P are collinear. Prove that AB \cdot CD = BC \cdot DA.

This is driving me CRAZY and the moment. mad
several swearing smileys should make me feel better. swearswearswear


Batman shows Superman his new phone. Batman-"Alfred, whats the temperature in London?"
Alfred-"Just a second, sir." Alfred turns on his iPhone."Siri, whats the temperature in London?"
Siri - "Six centigrade." Alfred-"Six centigrade sir." Batman-"See?"

#2 2014-02-09 21:53:50

bob bundy
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Re: sort of power of a point

HI thedarktiger,

I know the feeling.  I still have some problems posted over a year ago on my 'to-do' list.  I remain optimistic about them.

EVEN LATER EDIT:

I thought this was going to come out easily; but it hasn't.  Nevertheless I think the approach is the right one ... I've just got in a muddle with my similar triangles.


Your problem is a little like a theorem about intersecting chords so I've used a similar method.  I haven't used the tangents at all; whoops!I haven't changed my mind about this.  If you pick any two points (not diametrically opposite) you can construct tangents that will intersect at P.  As P doesn't feature in the required equation, I cannot see why you cannot treat P as any point just to determine B and C.  So all the 'action' should be able to take place inside the circle.  As I type this, I'm already thinking I'm wrong so maybe I'll have a new approach soon smile )

I have now explored what happens if P isn't as specified and the property fails to hold.  Conclusion: what I wrote earlier is wrong so I shall delete it and go back to the drawing board.


Bob


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Last edited by bob bundy (2014-02-09 22:41:18)


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

#3 2014-02-10 08:00:50

bob bundy
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Re: sort of power of a point

hi,

It has driven me crazy too.  I have drawn and re-drawn the diagram with varied emphasis and called angles x, y and z and then found every other angle in terms of these.  I have experimented with similar triangles until my head feels like bursting.  And still no joy.  sad

With my geometry program I have verified that it is, indeed, true.  But I am at a loss as to how to prove it.

Very sorry.  Maybe tomorrow ..........

Bob


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

#4 2014-02-10 16:52:53

anonimnystefy
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Re: sort of power of a point

Hi guys,

- look at your own responsibility smile

Last edited by anonimnystefy (2014-02-10 16:56:04)


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#5 2014-02-10 21:25:15

thedarktiger
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Re: sort of power of a point

Thanks for the anwser, anonimnystefy! I think my head has already burst. That problem was plain evil.
I'm sure anybody named bob will agree. aaaaaaggggghhhhhh.


Batman shows Superman his new phone. Batman-"Alfred, whats the temperature in London?"
Alfred-"Just a second, sir." Alfred turns on his iPhone."Siri, whats the temperature in London?"
Siri - "Six centigrade." Alfred-"Six centigrade sir." Batman-"See?"

#6 2014-02-11 01:19:39

anonimnystefy
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Re: sort of power of a point

You're welcome! smile

The problem is quite tricky!


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#7 2014-02-11 01:29:19

bob bundy
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Re: sort of power of a point

hi Stefy,

Thanks for putting me out of my misery.  I'll be able to sleep now. sleep

Bob


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

#8 2014-02-11 14:08:22

anonimnystefy
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Re: sort of power of a point

You're welcome, but I sure am hoping you are using that as a figure of speech only. smile


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#9 2014-02-11 19:40:50

thedarktiger
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Re: sort of power of a point

Yup. don't want a math problem to do THAT. big_smile


Batman shows Superman his new phone. Batman-"Alfred, whats the temperature in London?"
Alfred-"Just a second, sir." Alfred turns on his iPhone."Siri, whats the temperature in London?"
Siri - "Six centigrade." Alfred-"Six centigrade sir." Batman-"See?"

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